From 84f0689e7e73f06f52e61425ce00aa457ae98b88 Mon Sep 17 00:00:00 2001
From: Martin Goik <goik@hdm-stuttgart.de>
Date: Sat, 26 Sep 2015 13:08:46 +0200
Subject: [PATCH] Wrapping lottery related binomials into a static method

---
 Doc/Sd1/Ref/Screen/lotto-spiel77.jpg          | Bin 0 -> 169008 bytes
 Doc/Sd1/Ref/Screen/superenalotto.gif          | Bin 0 -> 45444 bytes
 Doc/Sd1/objectsClasses.xml                    | 476 ++++++++++++++
 Doc/Sd1/statements.xml                        | 611 +++++++++++++++++-
 P/Sd1/Binomial/V1/.gitignore                  |   5 +
 P/Sd1/Binomial/V1/pom.xml                     |  20 +
 .../mi/sd1/binomial/Binomial.java             |  32 +
 .../sd1/binomial/FactorialLongLimitTest.java  |  15 +
 .../mi/sd1/binomial/Lottery.java              |  18 +
 .../mi/sd1/binomial/test/BinomTest.java       |  60 ++
 10 files changed, 1228 insertions(+), 9 deletions(-)
 create mode 100644 Doc/Sd1/Ref/Screen/lotto-spiel77.jpg
 create mode 100644 Doc/Sd1/Ref/Screen/superenalotto.gif
 create mode 100644 P/Sd1/Binomial/V1/.gitignore
 create mode 100644 P/Sd1/Binomial/V1/pom.xml
 create mode 100644 P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Binomial.java
 create mode 100644 P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/FactorialLongLimitTest.java
 create mode 100644 P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Lottery.java
 create mode 100644 P/Sd1/Binomial/V1/src/test/java/de/hdm_stuttgart/mi/sd1/binomial/test/BinomTest.java

diff --git a/Doc/Sd1/Ref/Screen/lotto-spiel77.jpg b/Doc/Sd1/Ref/Screen/lotto-spiel77.jpg
new file mode 100644
index 0000000000000000000000000000000000000000..14a5ee848905e243d2511cf82c2bcac98ae5d29f
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diff --git a/Doc/Sd1/objectsClasses.xml b/Doc/Sd1/objectsClasses.xml
index bfba5544e..6130ddfa2 100644
--- a/Doc/Sd1/objectsClasses.xml
+++ b/Doc/Sd1/objectsClasses.xml
@@ -2126,6 +2126,482 @@ long sum = (long)a + b;</programlisting>
       </section>
     </section>
 
+    <section xml:id="sd1SectLotteryRevisited">
+      <title>Lotteries revisited</title>
+
+      <qandaset defaultlabel="qanda" xml:id="sd1QandaLotteryRevisited">
+        <qandadiv>
+          <qandaentry>
+            <question>
+              <para>Wrap your implementation from <xref
+              linkend="sd1SectPlayingLottery"/> into a single method
+              <methodname>binomial(int n, int k)</methodname>. Then manually
+              calculate smaller values such as <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>4</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>3</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> beforehand to define corresponding unit
+              tests. Do not forget: <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>1</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>0</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> and <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>0</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>0</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> are valid expressions as well.</para>
+
+              <para>Manually calculate <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>22</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>2</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> and <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>22</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>20</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> beforehand to add more unit tests. Your
+              implementation is likely to fail for (at least) one of these.
+              Why does this happen? May you propose a solution? Keep in mind
+              the idea to deal with <quote>6 out of 90</quote> lottery
+              situations.</para>
+
+              <tip>
+                <para>Be careful about possible arithmetic overflows.</para>
+              </tip>
+            </question>
+
+            <answer>
+              <para>A complete implementation among with some unit tests is
+              available at:</para>
+
+              <annotation role="make">
+                <para role="eclipse">Sd1/Binomial/V1</para>
+              </annotation>
+
+              <para>This actually wraps <xref
+              linkend="sd1EqnBinomialCalculate"/> into a method
+              <methodname>public static long binomial(int n, int
+              k)</methodname>. </para>
+
+              <para>What happens when calculating <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>22</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>21</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> ?</para>
+
+              <programlisting language="java">    // Numerator product n (n - 1) ... (n - k + 1)
+    long numerator = 1;
+    for (int i = n - k + 1; i &lt;= n; i++) {
+      numerator *= i;
+    }</programlisting>
+
+              <para>In this case our numerator variable of type
+              <code>long</code> will be set to a value 22! (factorial).
+              Consider a simple demo code snippet among with its
+              result:</para>
+
+              <informaltable border="1">
+                <colgroup width="10%"/>
+
+                <colgroup width="90%"/>
+
+                <tr>
+                  <th>Code</th>
+
+                  <td valign="top"><programlisting language="java">  public static void main(String[] args) {
+    long factorial = 1;
+    
+    for (int i = 2; i &lt; 22; i++) {
+      factorial *= i;
+      System.out.println(factorial);
+    }
+  }</programlisting></td>
+                </tr>
+
+                <tr>
+                  <th>Result</th>
+
+                  <td valign="top"><programlisting language="none">2: 2
+3: 6
+4: 24
+5: 120
+6: 720
+7: 5040
+8: 40320
+9: 362880
+10: 3628800
+11: 39916800
+12: 479001600
+13: 6227020800
+14: 87178291200
+15: 1307674368000
+16: 20922789888000
+17: 355687428096000
+18: 6402373705728000
+19: 121645100408832000
+20: 2432902008176640000
+21: -4249290049419214848
+
+Largest long value:9223372036854775807</programlisting></td>
+                </tr>
+              </informaltable>
+
+              <para>So 21! already yields a negative value. Actually
+              <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mi>20</m:mi>
+
+                        <m:mo>!</m:mo>
+                      </m:mrow>
+
+                      <m:mo>=</m:mo>
+
+                      <m:mi>2432902008176640000</m:mi>
+
+                      <m:mo>&lt;</m:mo>
+
+                      <m:mi>9223372036854775807</m:mi>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> still holds. The subsequent multiplication
+              by 21 however results in an arithmetic overflow.</para>
+
+              <para>This does not show up as an error since exactly the same
+              (mis)calculation happens for the denominator. Thus their
+              quotient by coincidence returns a correct value of 1. This
+              however is no longer true when examining <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>22</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>20</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation>.</para>
+
+              <para>We may still get correct results in many situations
+              including the given example. If k gets too large we may use:
+              </para>
+
+              <informalequation>
+                <m:math display="block">
+                  <m:mrow>
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>n</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>k</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+
+                    <m:mo>=</m:mo>
+
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>n</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mrow>
+                                  <m:mi>n</m:mi>
+
+                                  <m:mo>-</m:mo>
+
+                                  <m:mi>k</m:mi>
+                                </m:mrow>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:mrow>
+                </m:math>
+              </informalequation>
+
+              <para>So instead of calculating <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>22</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>20</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> we take its sibling <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>22</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>2</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation> having identical value by adding a simple
+              <code>if</code> statement:</para>
+
+              <programlisting language="java">  public static long binomial(int n, int k) {
+    
+    // Trying to avoid arithmetic overflow by making use of:
+    //             n        n
+    //           (   ) =  (   )
+    //             k       n-k
+    //
+    if (n / 2 &lt; k) {
+      k = n - k;
+    }
+    
+    // Numerator product n (n - 1) ... (n - k + 1)
+    long numerator = 1;
+ ...</programlisting>
+
+              <para>Finally our lottery code from <xref
+              linkend="sd1SectPlayingLottery"/> gets a little bit
+              simplified:</para>
+
+              <informaltable border="1">
+                <colgroup width="10%"/>
+
+                <colgroup width="90%"/>
+
+                <tr>
+                  <th>Code</th>
+
+                  <td valign="top"><programlisting language="java">    final int
+      totalNumberCount = 49,
+      drawnNumberCount = 6;
+    
+    System.out.println(
+        "Your chance to win when drawing " + drawnNumberCount + 
+        " out of " + totalNumberCount + 
+        " is 1 : " + <emphasis role="bold">Binomial.binomial(totalNumberCount, drawnNumberCount))</emphasis>;
+  }</programlisting></td>
+                </tr>
+
+                <tr>
+                  <th>Result</th>
+
+                  <td valign="top"><programlisting language="none">Your chance to win when drawing 6 out of 49 is 1 : 13983816</programlisting></td>
+                </tr>
+              </informaltable>
+            </answer>
+          </qandaentry>
+        </qandadiv>
+      </qandaset>
+    </section>
+
     <section xml:id="sd1Gcd">
       <title>The greatest common divisor and the common multiple</title>
 
diff --git a/Doc/Sd1/statements.xml b/Doc/Sd1/statements.xml
index 8bdddebba..390b8b4ef 100644
--- a/Doc/Sd1/statements.xml
+++ b/Doc/Sd1/statements.xml
@@ -347,9 +347,10 @@ Decimal value 14 not yet implemented</programlisting>
           <qandaentry>
             <question>
               <para>Write an application to print an ASCII art Xmas tree.
-              Provide a configurable parameter which allows for controlling
-              the tree's size. The following example shows a tree spanning 6
-              rows excluding top (X) and bottom (###):</para>
+              Provide a configurable parameter <code>numberOfRows</code> which
+              allows for controlling the tree's size. The following example
+              shows a tree spanning 6 rows excluding the top (X) and the two
+              bottom (###) lines:</para>
 
               <programlisting language="none">      X
       *
@@ -400,7 +401,7 @@ Decimal value 14 not yet implemented</programlisting>
             </question>
 
             <answer>
-              <programlisting language="none">   public static void main(String[] args) {      
+              <programlisting language="java">   public static void main(String[] args) {      
       
       // Example: 6 rows, tree's body loop index ranging from 0 to 5
       //
@@ -488,8 +489,8 @@ Decimal value 14 not yet implemented</programlisting>
             </question>
 
             <answer>
-              <para>We start by a version being fully covered by our knowledge
-              gained so far:</para>
+              <para>We start from a version being fully covered by our current
+              knowledge:</para>
 
               <programlisting language="java">   public static void main(String[] args) {      
       
@@ -1209,6 +1210,598 @@ Decimal value 14 not yet implemented</programlisting>
       </section>
     </section>
 
+    <section xml:id="sd1SectPlayingLottery">
+      <title>Playing lottery</title>
+
+      <qandaset defaultlabel="qanda" xml:id="sd1QandaPlayingLottery">
+        <qandadiv>
+          <qandaentry>
+            <question>
+              <para>Many lotteries randomly dram numbers from a given set
+              like:</para>
+
+              <glosslist>
+                <glossentry>
+                  <glossterm>Germany / <quote
+                  xml:lang="de">Glücksspirale</quote></glossterm>
+
+                  <glossdef>
+                    <informaltable border="1">
+                      <colgroup width="14%"/>
+
+                      <colgroup width="86%"/>
+
+                      <tr>
+                        <td valign="top">6 out of 49</td>
+
+                        <td><mediaobject>
+                            <imageobject>
+                              <imagedata fileref="Ref/Screen/lotto-spiel77.jpg"/>
+                            </imageobject>
+                          </mediaobject></td>
+                      </tr>
+                    </informaltable>
+                  </glossdef>
+                </glossentry>
+
+                <glossentry>
+                  <glossterm>Italy / <quote>SuperEnalotto</quote></glossterm>
+
+                  <glossdef>
+                    <informaltable border="1">
+                      <colgroup width="14%"/>
+
+                      <colgroup width="86%"/>
+
+                      <tr>
+                        <td valign="top">6 out of 90</td>
+
+                        <td><mediaobject>
+                            <imageobject>
+                              <imagedata fileref="Ref/Screen/superenalotto.gif"/>
+                            </imageobject>
+                          </mediaobject></td>
+                      </tr>
+                    </informaltable>
+                  </glossdef>
+                </glossentry>
+              </glosslist>
+
+              <para>This category of lottery does not care about ordering
+              (apart from so called jolly numbers). The number of
+              possibilities drawing n out of m numbers ignoring their order is
+              being given by the <link
+              xlink:href="https://en.wikipedia.org/wiki/Binomial_coefficient">binomial
+              coefficient <inlineequation>
+                  <m:math display="inline">
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>n</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>k</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+                  </m:math>
+                </inlineequation></link>:</para>
+
+              <informalequation>
+                <m:math display="block">
+                  <m:mrow>
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>n</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>k</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+
+                    <m:mo>=</m:mo>
+
+                    <m:mfrac>
+                      <m:mrow>
+                        <m:mi>n</m:mi>
+
+                        <m:mo>!</m:mo>
+                      </m:mrow>
+
+                      <m:mrow>
+                        <m:mrow>
+                          <m:mi>k</m:mi>
+
+                          <m:mo>!</m:mo>
+                        </m:mrow>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mrow>
+                          <m:mrow>
+                            <m:mo>(</m:mo>
+
+                            <m:mrow>
+                              <m:mi>n</m:mi>
+
+                              <m:mo>-</m:mo>
+
+                              <m:mi>k</m:mi>
+                            </m:mrow>
+
+                            <m:mo>)</m:mo>
+                          </m:mrow>
+
+                          <m:mo>!</m:mo>
+                        </m:mrow>
+                      </m:mrow>
+                    </m:mfrac>
+                  </m:mrow>
+                </m:math>
+              </informalequation>
+
+              <para>Write an application which allows for calculation of
+              probabilistic success rates. For the German <quote
+              xml:lang="de">Glücksspirale</quote> a possible output
+              reads:</para>
+
+              <programlisting language="none">Your chance to win when drawing 6 out of 49 is 1 : 13983816</programlisting>
+
+              <tip>
+                <orderedlist>
+                  <listitem>
+                    <para>Store parameters like 6 or 49 in variable to keep
+                    your software flexible.</para>
+                  </listitem>
+
+                  <listitem>
+                    <para>Use <code>long</code> variables when appropriate to
+                    avoid overflow errors.</para>
+                  </listitem>
+
+                  <listitem>
+                    <para>Even if using variables of type <code>long</code>
+                    you will not be able to represent e.g. <quote>90!</quote>
+                    due to an arithmetic overflow. This problem can be avoided
+                    by simplifying <inlineequation>
+                        <m:math display="inline">
+                          <m:mfrac>
+                            <m:mrow>
+                              <m:mi>n</m:mi>
+
+                              <m:mo>!</m:mo>
+                            </m:mrow>
+
+                            <m:mrow>
+                              <m:mrow>
+                                <m:mi>k</m:mi>
+
+                                <m:mo>!</m:mo>
+                              </m:mrow>
+
+                              <m:mo>⁢</m:mo>
+
+                              <m:mrow>
+                                <m:mrow>
+                                  <m:mo>(</m:mo>
+
+                                  <m:mrow>
+                                    <m:mi>n</m:mi>
+
+                                    <m:mo>-</m:mo>
+
+                                    <m:mi>k</m:mi>
+                                  </m:mrow>
+
+                                  <m:mo>)</m:mo>
+                                </m:mrow>
+
+                                <m:mo>!</m:mo>
+                              </m:mrow>
+                            </m:mrow>
+                          </m:mfrac>
+                        </m:math>
+                      </inlineequation> beforehand.</para>
+                  </listitem>
+
+                  <listitem>
+                    <para>The following loop allows for computing a given
+                    number's factorial:</para>
+
+                    <programlisting language="java">long factorial = 1;
+for (int i = 2; i &lt;= number; i++) {
+  factorial *= i;
+}</programlisting>
+
+                    <para>Use similar loops to solve the problem.</para>
+                  </listitem>
+                </orderedlist>
+              </tip>
+            </question>
+
+            <answer>
+              <para>By simple fraction cancellation we get:</para>
+
+              <equation xml:id="sd1EqnBinomialCalculate">
+                <title>Calculating binomials in a slightly more efficient
+                manner rather than just computing three factorials.</title>
+
+                <m:math display="block">
+                  <m:mrow>
+                    <m:mfrac>
+                      <m:mrow>
+                        <m:mi>n</m:mi>
+
+                        <m:mo>!</m:mo>
+                      </m:mrow>
+
+                      <m:mrow>
+                        <m:mrow>
+                          <m:mi>k</m:mi>
+
+                          <m:mo>!</m:mo>
+                        </m:mrow>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mrow>
+                          <m:mrow>
+                            <m:mo>(</m:mo>
+
+                            <m:mrow>
+                              <m:mi>n</m:mi>
+
+                              <m:mo>-</m:mo>
+
+                              <m:mi>k</m:mi>
+                            </m:mrow>
+
+                            <m:mo>)</m:mo>
+                          </m:mrow>
+
+                          <m:mo>!</m:mo>
+                        </m:mrow>
+                      </m:mrow>
+                    </m:mfrac>
+
+                    <m:mo>=</m:mo>
+
+                    <m:mfrac>
+                      <m:mrow>
+                        <m:mi>n</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mrow>
+                          <m:mo>(</m:mo>
+
+                          <m:mrow>
+                            <m:mi>n</m:mi>
+
+                            <m:mo>-</m:mo>
+
+                            <m:mi>1</m:mi>
+                          </m:mrow>
+
+                          <m:mo>)</m:mo>
+                        </m:mrow>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>...</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>1</m:mi>
+                      </m:mrow>
+
+                      <m:mrow>
+                        <m:mrow>
+                          <m:mi>k</m:mi>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mrow>
+                            <m:mo>(</m:mo>
+
+                            <m:mrow>
+                              <m:mi>k</m:mi>
+
+                              <m:mo>-</m:mo>
+
+                              <m:mi>1</m:mi>
+                            </m:mrow>
+
+                            <m:mo>)</m:mo>
+                          </m:mrow>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mi>...</m:mi>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mi>1</m:mi>
+                        </m:mrow>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mrow>
+                          <m:mrow>
+                            <m:mo>(</m:mo>
+
+                            <m:mrow>
+                              <m:mi>n</m:mi>
+
+                              <m:mo>-</m:mo>
+
+                              <m:mi>k</m:mi>
+                            </m:mrow>
+
+                            <m:mo>)</m:mo>
+                          </m:mrow>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mrow>
+                            <m:mo>(</m:mo>
+
+                            <m:mrow>
+                              <m:mi>n</m:mi>
+
+                              <m:mo>-</m:mo>
+
+                              <m:mi>k</m:mi>
+
+                              <m:mo>-</m:mo>
+
+                              <m:mi>1</m:mi>
+                            </m:mrow>
+
+                            <m:mo>)</m:mo>
+                          </m:mrow>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mi>...</m:mi>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mi>1</m:mi>
+                        </m:mrow>
+                      </m:mrow>
+                    </m:mfrac>
+
+                    <m:mo>=</m:mo>
+
+                    <m:mfrac>
+                      <m:mrow>
+                        <m:mi>n</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mrow>
+                          <m:mo>(</m:mo>
+
+                          <m:mrow>
+                            <m:mi>n</m:mi>
+
+                            <m:mo>-</m:mo>
+
+                            <m:mi>1</m:mi>
+                          </m:mrow>
+
+                          <m:mo>)</m:mo>
+                        </m:mrow>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>...</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mrow>
+                          <m:mo>(</m:mo>
+
+                          <m:mrow>
+                            <m:mi>n</m:mi>
+
+                            <m:mo>-</m:mo>
+
+                            <m:mi>k</m:mi>
+
+                            <m:mo>+</m:mo>
+
+                            <m:mi>1</m:mi>
+                          </m:mrow>
+
+                          <m:mo>)</m:mo>
+                        </m:mrow>
+                      </m:mrow>
+
+                      <m:mrow>
+                        <m:mrow>
+                          <m:mi>k</m:mi>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mrow>
+                            <m:mo>(</m:mo>
+
+                            <m:mrow>
+                              <m:mi>k</m:mi>
+
+                              <m:mo>-</m:mo>
+
+                              <m:mi>1</m:mi>
+                            </m:mrow>
+
+                            <m:mo>)</m:mo>
+                          </m:mrow>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mi>...</m:mi>
+
+                          <m:mo>⁢</m:mo>
+
+                          <m:mi>1</m:mi>
+                        </m:mrow>
+                      </m:mrow>
+                    </m:mfrac>
+                  </m:mrow>
+                </m:math>
+              </equation>
+
+              <para>As an example we have:</para>
+
+              <informalequation>
+                <m:math display="block">
+                  <m:mrow>
+                    <m:mrow>
+                      <m:mrow>
+                        <m:mo>(</m:mo>
+
+                        <m:mrow>
+                          <m:mtable>
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>49</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+
+                            <m:mtr>
+                              <m:mtd>
+                                <m:mi>6</m:mi>
+                              </m:mtd>
+                            </m:mtr>
+                          </m:mtable>
+                        </m:mrow>
+
+                        <m:mo>)</m:mo>
+                      </m:mrow>
+                    </m:mrow>
+
+                    <m:mo>=</m:mo>
+
+                    <m:mfrac>
+                      <m:mrow>
+                        <m:mi>49</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>48</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>47</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>46</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>45</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>44</m:mi>
+                      </m:mrow>
+
+                      <m:mrow>
+                        <m:mi>6</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>5</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>4</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>3</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>2</m:mi>
+
+                        <m:mo>⁢</m:mo>
+
+                        <m:mi>1</m:mi>
+                      </m:mrow>
+                    </m:mfrac>
+                  </m:mrow>
+                </m:math>
+              </informalequation>
+
+              <para>Calculating numerator and denominator in two separate
+              loops a possible implementation reads:</para>
+
+              <programlisting language="java">public class Lottery {
+
+  public static void main(String[] args) {
+
+    // Input values:
+    final int
+      totalNumberCount = 49,
+      drawnNumberCount = 6;
+    
+    // No changes below this line
+
+    
+    // Numerator loop
+    long numerator = 1;
+    for (int i = totalNumberCount - drawnNumberCount + 1; i &lt;= totalNumberCount; i++) {
+      numerator *= i;
+    }
+    
+    // Denominator loop calculating the "smaller" factorial
+    long factorial = 1;
+    for (int i = 2; i &lt;= drawnNumberCount; i++) {
+      factorial *= i;
+    }
+
+    System.out.println("Your chance to win when drawing " + drawnNumberCount + 
+        " out of " + totalNumberCount + " is 1 : " + (numerator / factorial));
+  }
+}</programlisting>
+            </answer>
+          </qandaentry>
+        </qandadiv>
+      </qandaset>
+    </section>
+
     <section xml:id="sd1SectNumberguess">
       <title>Guessing numbers</title>
 
@@ -2058,17 +2651,17 @@ Congratulations! you won!</programlisting>
       <title>Pythagorean triples</title>
 
       <qandaset defaultlabel="qanda" xml:id="sw1QandaPythagoreanTriples">
-        <title>Finding pythagorean triples</title>
+        <title>Finding Pythagorean triples</title>
 
         <qandadiv>
           <qandaentry>
             <question>
               <para>Read the <link
               xlink:href="http://en.wikipedia.org/wiki/Pythagorean_triple">definition
-              of pythagorean triples</link>.</para>
+              of Pythagorean triples</link>.</para>
 
               <para>Find all <link
-              xlink:href="http://en.wikipedia.org/wiki/Pythagorean_triple">pythagorean
+              xlink:href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean
               triples</link> (a,b,c) for which which in addition to their
               definition the restriction <inlineequation>
                   <m:math display="inline">
diff --git a/P/Sd1/Binomial/V1/.gitignore b/P/Sd1/Binomial/V1/.gitignore
new file mode 100644
index 000000000..cf5fd6bb2
--- /dev/null
+++ b/P/Sd1/Binomial/V1/.gitignore
@@ -0,0 +1,5 @@
+/.settings
+/target
+/.classpath
+/.project
+/A1.log
diff --git a/P/Sd1/Binomial/V1/pom.xml b/P/Sd1/Binomial/V1/pom.xml
new file mode 100644
index 000000000..ecd1d0edd
--- /dev/null
+++ b/P/Sd1/Binomial/V1/pom.xml
@@ -0,0 +1,20 @@
+<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
+  xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/xsd/maven-4.0.0.xsd">
+  <modelVersion>4.0.0</modelVersion>
+
+  <parent>
+    <groupId>de.hdm-stuttgart.mi</groupId>
+    <artifactId>lecturenotes-pom</artifactId>
+    <version>1.0</version>
+    <relativePath>../../../pom.xml</relativePath>
+  </parent>
+
+  <groupId>de.hdm-stuttgart.de.sd1</groupId>
+  <artifactId>binomial</artifactId>
+  <version>1.1</version>
+  <packaging>jar</packaging>
+
+  <name>Binomial</name>
+  <url>http://www.mi.hdm-stuttgart.de/freedocs</url>
+
+</project>
diff --git a/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Binomial.java b/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Binomial.java
new file mode 100644
index 000000000..e91f42148
--- /dev/null
+++ b/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Binomial.java
@@ -0,0 +1,32 @@
+package de.hdm_stuttgart.mi.sd1.binomial;
+
+public class Binomial {
+
+  public static long binomial(int n, int k) {
+    
+    // Trying to avoid arithmetic overflow by making use of:
+    //             n        n
+    //           (   ) =  (   )
+    //             k       n-k
+    //
+    if (n / 2 < k) {
+      k = n - k;
+    }
+    
+    // Numerator product n (n - 1) ... (n - k + 1)
+    long numerator = 1;
+    for (int i = n - k + 1; i <= n; i++) {
+      numerator *= i;
+    }
+    return numerator / factorial(k);
+  }
+  
+  public static long factorial(int n) {
+    long ret = 1;
+    
+    for (int i = 2; i <= n; i++) {
+      ret *= i;
+    }
+    return ret;
+  }
+}
diff --git a/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/FactorialLongLimitTest.java b/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/FactorialLongLimitTest.java
new file mode 100644
index 000000000..c74094e6f
--- /dev/null
+++ b/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/FactorialLongLimitTest.java
@@ -0,0 +1,15 @@
+package de.hdm_stuttgart.mi.sd1.binomial;
+
+public class FactorialLongLimitTest {
+
+  public static void main(String[] args) {
+    long factorial = 1;
+    
+    for (int i = 2; i < 22; i++) {
+      factorial *= i;
+      System.out.println( i + ": " + factorial);
+    }
+    
+    System.out.println("\nLargest long value:" + Long.MAX_VALUE);
+  }
+}
diff --git a/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Lottery.java b/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Lottery.java
new file mode 100644
index 000000000..7a4dd57f1
--- /dev/null
+++ b/P/Sd1/Binomial/V1/src/main/java/de/hdm_stuttgart/mi/sd1/binomial/Lottery.java
@@ -0,0 +1,18 @@
+package de.hdm_stuttgart.mi.sd1.binomial;
+
+public class Lottery {
+
+  public static void main(String[] args) {
+
+    // Input values:
+    final int
+      totalNumberCount = 49,
+      drawnNumberCount = 6;
+    
+    System.out.println(
+        "Your chance to win when drawing " + drawnNumberCount + 
+        " out of " + totalNumberCount + 
+        " is 1 : " + Binomial.binomial(totalNumberCount, drawnNumberCount));
+  }
+  
+}
diff --git a/P/Sd1/Binomial/V1/src/test/java/de/hdm_stuttgart/mi/sd1/binomial/test/BinomTest.java b/P/Sd1/Binomial/V1/src/test/java/de/hdm_stuttgart/mi/sd1/binomial/test/BinomTest.java
new file mode 100644
index 000000000..fa8c806ab
--- /dev/null
+++ b/P/Sd1/Binomial/V1/src/test/java/de/hdm_stuttgart/mi/sd1/binomial/test/BinomTest.java
@@ -0,0 +1,60 @@
+package de.hdm_stuttgart.mi.sd1.binomial.test;
+
+import org.junit.Assert;
+import org.junit.Test;
+
+import de.hdm_stuttgart.mi.sd1.binomial.Binomial;
+
+public class BinomTest {
+
+  @Test
+  public void test_0_0() {
+    Assert.assertEquals(1, Binomial.binomial(0, 0));
+  }
+  
+  @Test
+  public void test_1_0() {
+    Assert.assertEquals(1, Binomial.binomial(1, 0));
+  }
+  
+  @Test
+  public void test_1_1() {
+    Assert.assertEquals(1, Binomial.binomial(1, 1));
+  }
+  
+  @Test
+  public void test_4_3() {
+    Assert.assertEquals(4, Binomial.binomial(4, 3));
+  }
+  
+  @Test
+  public void test_5_3() {
+    Assert.assertEquals(10, Binomial.binomial(5, 3));
+  }
+  
+  @Test
+  public void test_20_20() {
+    Assert.assertEquals(1, Binomial.binomial(20, 20));
+  }
+
+  // Possible arithmetic overflow situations.
+  @Test
+  public void test_22_1() {
+    Assert.assertEquals(22, Binomial.binomial(22, 1));
+  }
+
+  @Test
+  public void test_22_21() {
+    Assert.assertEquals(22, Binomial.binomial(22, 21));
+  }
+
+  @Test
+  public void test_22_2() {
+    Assert.assertEquals(231, Binomial.binomial(22, 2));
+  }
+
+  @Test
+  public void test_22_20() {
+    Assert.assertEquals(231, Binomial.binomial(22, 20));
+  }
+}
-- 
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