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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.factorials.sf_binomial"></a><a class="link" href="sf_binomial.html" title="Binomial Coefficients">Binomial Coefficients</a>
  28 </h3></div></div></div>
  29 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  30 </pre>
  31 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  34 <span class="identifier">T</span> <span class="identifier">binomial_coefficient</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">k</span><span class="special">);</span>
  35
  36 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  37 <span class="identifier">T</span> <span class="identifier">binomial_coefficient</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  38
  39 <span class="special">}}</span> <span class="comment">// namespaces</span>
  40 </pre>
  41 <p>
  42         Returns the binomial coefficient: <sub>n</sub>C<sub>k</sub>.
  43       </p>
  44 <p>
  45         Requires k &lt;= n.
  46       </p>
  47 <p>
  48         The final <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  49         be used to control the behaviour of the function: how it handles errors,
  50         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">policy
  51         documentation for more details</a>.
  52       </p>
  53 <p>
  54         May return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
  55         if the result is too large to represent in type T.
  56       </p>
  57 <div class="important"><table border="0" summary="Important">
  58 <tr>
  59 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Important]" src="../../../../../../doc/src/images/important.png"></td>
  60 <th align="left">Important</th>
  61 </tr>
  62 <tr><td align="left" valign="top">
  63 <p>
  64           The functions described above are templates where the template argument
  65           T can not be deduced from the arguments passed to the function. Therefore
  66           if you write something like:
  67         </p>
  68 <p>
  69           <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial_coefficient</span><span class="special">(</span><span class="number">10</span><span class="special">,</span> <span class="number">2</span><span class="special">);</span></code>
  70         </p>
  71 <p>
  72           You will get a compiler error, ususally indicating that there is no such
  73           function to be found. Instead you need to specifiy the return type explicity
  74           and write:
  75         </p>
  76 <p>
  77           <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial_coefficient</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">10</span><span class="special">,</span> <span class="number">2</span><span class="special">);</span></code>
  78         </p>
  79 <p>
  80           So that the return type is known. Further, the template argument must be
  81           a real-valued type such as <code class="computeroutput"><span class="keyword">float</span></code>
  82           or <code class="computeroutput"><span class="keyword">double</span></code> and not an integer
  83           type - that would overflow far too easily!
  84         </p>
  85 </td></tr>
  86 </table></div>
  87 <h5>
  88 <a name="math_toolkit.factorials.sf_binomial.h0"></a>
  89         <span class="phrase"><a name="math_toolkit.factorials.sf_binomial.accuracy"></a></span><a class="link" href="sf_binomial.html#math_toolkit.factorials.sf_binomial.accuracy">Accuracy</a>
  90       </h5>
  91 <p>
  92         The accuracy will be the same as for the factorials for small arguments (i.e.
  93         no more than one or two epsilon), and the <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>
  94         function for larger arguments.
  95       </p>
  96 <h5>
  97 <a name="math_toolkit.factorials.sf_binomial.h1"></a>
  98         <span class="phrase"><a name="math_toolkit.factorials.sf_binomial.testing"></a></span><a class="link" href="sf_binomial.html#math_toolkit.factorials.sf_binomial.testing">Testing</a>
  99       </h5>
 100 <p>
 101         The spot tests for the binomial coefficients use data generated by functions.wolfram.com.
 102       </p>
 103 <h5>
 104 <a name="math_toolkit.factorials.sf_binomial.h2"></a>
 105         <span class="phrase"><a name="math_toolkit.factorials.sf_binomial.implementation"></a></span><a class="link" href="sf_binomial.html#math_toolkit.factorials.sf_binomial.implementation">Implementation</a>
 106       </h5>
 107 <p>
 108         Binomial coefficients are calculated using table lookup of factorials where
 109         possible using:
 110       </p>
 111 <p>
 112         <sub>n</sub>C<sub>k</sub> = n! / (k!(n-k)!)
 113       </p>
 114 <p>
 115         Otherwise it is implemented in terms of the beta function using the relations:
 116       </p>
 117 <p>
 118         <sub>n</sub>C<sub>k</sub> = 1 / (k * <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>(k,
 119         n-k+1))
 120       </p>
 121 <p>
 122         and
 123       </p>
 124 <p>
 125         <sub>n</sub>C<sub>k</sub> = 1 / ((n-k) * <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>(k+1,
 126         n-k))
 127       </p>
 128 </div>
 129 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 130 <td align="left"></td>
 131 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 132       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 133       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
 134       Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 135         Distributed under the Boost Software License, Version 1.0. (See accompanying
 136         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 137       </p>
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