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96 lines
4.3 KiB
HTML
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<!-- This manual describes how to install and use the GNU multiple precision
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arithmetic library, version 6.1.0.
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Copyright 1991, 1993-2015 Free Software Foundation, Inc.
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<meta name="description" content="How to install and use the GNU multiple precision arithmetic library, version 6.1.0.">
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<link href="Fibonacci-Numbers-Algorithm.html#Fibonacci-Numbers-Algorithm" rel="next" title="Fibonacci Numbers Algorithm">
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<a name="Binomial-Coefficients-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Fibonacci-Numbers-Algorithm.html#Fibonacci-Numbers-Algorithm" accesskey="n" rel="next">Fibonacci Numbers Algorithm</a>, Previous: <a href="Factorial-Algorithm.html#Factorial-Algorithm" accesskey="p" rel="prev">Factorial Algorithm</a>, Up: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="u" rel="up">Other Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
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<a name="Binomial-Coefficients"></a>
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<h4 class="subsection">15.7.3 Binomial Coefficients</h4>
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<a name="index-Binomial-coefficient-algorithm"></a>
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<p>Binomial coefficients <em>C(n,k)</em> are calculated
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by first arranging <em>k <= n/2</em> using <em>C(n,k) = C(n,n-k)</em> if necessary, and then
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evaluating the following product simply from <em>i=2</em> to <em>i=k</em>.
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</p>
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<div class="example">
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<pre class="example"> k (n-k+i)
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C(n,k) = (n-k+1) * prod -------
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i=2 i
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</pre></div>
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<p>It’s easy to show that each denominator <em>i</em> will divide the product so
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far, so the exact division algorithm is used (see <a href="Exact-Division.html#Exact-Division">Exact Division</a>).
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</p>
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<p>The numerators <em>n-k+i</em> and denominators <em>i</em> are first accumulated
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into as many fit a limb, to save multi-precision operations, though for
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<code>mpz_bin_ui</code> this applies only to the divisors, since <em>n</em> is an
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<code>mpz_t</code> and <em>n-k+i</em> in general won’t fit in a limb at all.
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</p>
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