<?xml version="1.0" encoding="UTF-8"?>
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 * Scilab ( https://www.scilab.org/ ) - This file is part of Scilab
 * Copyright (C) 2009 - 2010 - DIGITEO - Michael Baudin
 * Copyright (C) 2012 - Michael Baudin
 * Copyright (C) 2019 - Samuel GOUGEON - Le Mans Université
 *
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 * pursuant to article 5.3.4 of the CeCILL v.2.1.
 * This file was originally licensed under the terms of the CeCILL v2.1,
 * and continues to be available under such terms.
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 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
          xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml"
          xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook"
          xmlns:scilab="http://www.scilab.org" xml:id="nchoosek" xml:lang="en">
  <refnamediv>
    <refname>nchoosek</refname>
    <refpurpose>
        Computes binomial numbers (n,k) = numbers of combinations
    </refpurpose>
  </refnamediv>

<refsynopsisdiv>
   <title>Syntax</title>
   <synopsis>
    b = nchoosek(n, k)
    logb = nchoosek(n, k, logFormat)
    [logb, b] = nchoosek(n, k, logFormat)
    [logb, b] = nchoosek(n, k)
   </synopsis>
</refsynopsisdiv>

<refsection>
    <title>Arguments</title>
    <variablelist>
        <varlistentry>
            <term>n, k</term>
            <listitem>
                arrays of positive decimal integers. If both <varname>n</varname> and
                <varname>k</varname> are not scalars, they must have the same size.
            </listitem>
        </varlistentry>
        <varlistentry>
            <term>b</term>
            <listitem>
                array of positive decimal integers, with the size of the biggest array
                <varname>n</varname> or <varname>k</varname> :
                C<subscript>n</subscript><superscript>k</superscript>
            </listitem>
        </varlistentry>
        <varlistentry>
            <term>logb</term>
            <listitem>
                array of positive decimal numbers, with the size of <varname>b</varname>:
                <literal>log10(C<subscript>n</subscript><superscript>k</superscript>)</literal>
            </listitem>
        </varlistentry>
        <varlistentry>
            <term>logFormat</term>
            <listitem>
                single word "log10" | "10.mant". "log10" by default.
                <itemizedlist>
                    <listitem>
                        If "log10" is used, <varname>logb</varname> returns <literal>log10(b)</literal>.
                        <para/>
                    </listitem>
                    <listitem>
                        If "10.mant" is used, then <varname>logb</varname> returns
                        <literal>int(log10(b)) + 10^(log10(b)-int(log10(b)))/10</literal>:
                        The 10-exponent of b is the logb integer part, while its fractional part
                        directly shows the mantissa/10 of b, in [1.0, 10)/10.
                    </listitem>
                </itemizedlist>
            </listitem>
        </varlistentry>
    </variablelist>
</refsection>

<refsection>
    <title>Description</title>
    <para>
        For every (n(i),k(i)) element-wise pair, nchoosek() computes the number of
        k<subscript>i</subscript>-element subsets of an n<subscript>i</subscript>-element set.
        It is mathematically defined by
        <latex alt="C(n,k) = n! / [ k! (n-k)! ]" fontsize="18">C_n^k = {{n!} \over {k! \,(n-k)!}}
        = {{(n-k+1)\cdot ..(n-1)\cdot n} \over {1\cdot 2\cdot ...(k-1)\cdot k}}</latex>.
        The implementation, though, uses the βeta function.
    </para>
    <para>
        If n &lt; 0, or k &lt; 0, or k &gt; n, then an error is generated.
    </para>
    <refsect3>
        <title>Note about floating point accuracy</title>
        <para>
            Despite n! overflows for n>170, nchoosek(n,k) is computed within almost the %eps relative
            accuracy for n much beyond 170.
        </para>
        <para>
            For any given n value, we know that C<subscript>n</subscript><superscript>k</superscript>
            is maximal for k=n/2. Scilab computes b = nchoosek(n,n/2) close to the %eps relative
            accuracy for n up to 1029. Beyond this value, b=%inf is returned.
        </para>
        <para></para>
        <para>
            This narrow n=1029 limit can be overcome by computing
            log10(C<subscript>n</subscript><superscript>k</superscript>) and returning
            it through the second output argument logb. This allows to use still bigger n values,
            and to get some valid information about the exponent and the mantissa of the true result.
        </para>
        <para>
            However, we must keep in mind that beyond n = 1/%eps ~ 4.10<superscript>15</superscript>,
            the numerical step between consecutive integers m and m+1 stored as floating point
            numbers become > 1. Then <literal>n*(n-1)</literal> is numerically equal to
            <literal>n*n</literal>, and getting reliable results becomes harder.
       </para>
        <para>
            The integer part of logb represents the exponent (of 10) of
            C<subscript>n</subscript><superscript>k</superscript>, whereas its fractional part
            is the log10 of its mantissa. When the integer part of logb increases, the more digits
            it takes among the 16 available ones in floating point encoding, the less remain
            to encode the mantissa. Then,
            knowing the mantissa of C<subscript>n</subscript><superscript>k</superscript> with
            at least one significant digit requires
            C<subscript>n</subscript><superscript>k</superscript>
            &lt; 10<superscript>10<superscript>14</superscript></superscript>.
       </para>
       <para>
            <emphasis role="bold">Relative accuracy on logb</emphasis>: As a rule of thumb,
            except for special k cases, the relative uncertainty on logb is of the order of
            %eps * n / k.
       </para>
    </refsect3>
</refsection>

<refsection>
   <title>Examples</title>
   <para>
        Simple examples
   </para>
   <programlisting role="example"><![CDATA[
c = nchoosek(4, 1)
c = nchoosek(5, 0)
c = nchoosek([4 5], [1 0])
   ]]></programlisting>
   <screen><![CDATA[
--> c = nchoosek(4, 1)
 c  =
   4.

--> c = nchoosek(5, 0)
 c  =
   1.

--> c = nchoosek([4 5], [1 0])
 c  =
   4.   1.
]]></screen>
   <programlisting role="example"><![CDATA[
nchoosek(10, 0:10)
nchoosek(4:12, 4)
   ]]></programlisting>
   <screen><![CDATA[
--> nchoosek(10, 0:10)
 ans  =
   1.  10.  45.  120.  210.  252.  210.  120.  45.  10.  1.

--> nchoosek(4:12, 4)
 ans  =
   1.  5.  15.  35.  70.  126.  210.  330.  495.
]]></screen>
    <para>
        For big values:
    </para>
   <programlisting role="example"><![CDATA[
exact = 2.050083865033972676e307;
c = nchoosek(10000, 134)
relerror = abs(c-exact)/exact
   ]]></programlisting>
   <screen><![CDATA[
--> c = nchoosek(10000, 134)
 c  =
   2.05D+307

--> relerror = abs(c-exact)/exact
 relerror  =
   7.959D-14
]]></screen>
    <para>
        The exact result for
        <literal>C<subscript>n</subscript><superscript>k</superscript>(n, 2)</literal> is
        <literal>n.(n-1)/2</literal>. Now, for values <literal>n > 1/%eps = 4e15</literal>,
        <literal>(n-1)</literal> is numerically identical to <literal>n</literal>. In no way
        we can expect an exact result below, but rather <literal>n^2/2</literal> :
    </para>
   <programlisting role="example"><![CDATA[
n = 1e20;
c = nchoosek(n, 2)
c / (n*n/2) - 1    // Relative error wrt the expected numerical value
   ]]></programlisting>
   <screen><![CDATA[
--> c = nchoosek(n, 2)
 c  =
   5.000D+39

--> c /(n*n/2) - 1
 ans  =
  -6.661D-15
]]></screen>
    <para>
    In logarithmic formats:
    </para>
   <programlisting role="example"><![CDATA[
[logb, b] = nchoosek(10000, 1234);  [b, logb]
logb = nchoosek(10000, 1234, "10.mant")

[logb, b] = nchoosek(1000, 500);
logb2 = nchoosek(1000, 500, "10.mant");
[b, logb, logb2]
logb = nchoosek(1000, 500, "log10")
   ]]></programlisting>
   <screen><![CDATA[
--> [logb, b] = nchoosek(10000, 1234);  [b, logb]
 ans  =
   Inf   1620.803261

--> logb = nchoosek(10000, 1234, "10.mant")
 logb  =
   1620.635713      // 6.35713D+1620

--> [logb, b] = nchoosek(1000, 500);
--> logb2 = nchoosek(1000, 500, "10.mant");
--> [b, logb, logb2]
 ans  =
   2.7029D+299   299.4318272   299.2702882

--> logb = nchoosek(1000, 500, "log10")
 logb  =
   299.4318272
]]></screen>
   <para>
        Drawing <literal>nchoosek(n,k)</literal> on the main floating point domain, up to
        10<superscript>300</superscript> :
   </para>
   <programlisting role="example"><![CDATA[
function ax= drawCnk(n)
    // Preparing data
    [N,K] = meshgrid(n);
    noOp = K>N;
    K(noOp) = 0;
    [logC, C] = nchoosek(N, K);
    C(noOp) = %nan;
    if max(n)<2000, logC = log10(C), end

    // Surface of Cnk data
    surf(N, K, logC);
    gce().color_mode = -2; // hidding the mesh
    plot2d([0 n],[0 n/2]);
    gce().children.data(:,3) = max(logC(logC<>%inf))

    // Axes settings
    xgrid(25,1,7)
    ax = gca();
    set(ax, "view","2d", "tight_limits",["on" "on" "off"], "grid_position","foreground");
    xtitle("","","","");
    ax.margins(2) = 0.05;

    // Color bar
    colorbar();
    cb = gce();
    cb.y_ticks.labels = msprintf("$\\Large 10^{%s}$\n", cb.y_ticks.labels);
    title(cb,"$C_n^k$", "fontsize",3)
endfunction


clf
drawlater

// Figure settings
f = gcf();
f.color_map = jet(100);
//f.axes_size = [840 570];

// Main plot
ax = drawCnk(0:10:1500); sca(ax);
xtitle("nchoosek(n, k)","n","k","$C_n^k$");
set([ax.title ax.x_label ax.y_label ax.z_label], "font_size",4);
xstring(1250, 450, "Overflow")
gce().font_size = 4;

// Inset
xsetech([0.11 0.11 0.37 0.42]);
ax = drawCnk(0:106);
ax.sub_ticks = [3 3];
gce().sub_ticks(2) = 4;

drawnow
   ]]></programlisting>
   <scilab:image><![CDATA[
function ax= drawCnk(n)
    // Preparing data
    [N,K] = meshgrid(n);
    noOp = K>N;
    K(noOp) = 0;
    [logC, C] = nchoosek(N, K);
    C(noOp) = %nan;
    if max(n)<2000, logC = log10(C), end

    // Surface of Cnk data
    surf(N, K, logC);
    gce().color_mode = -2; // hidding the mesh
    plot2d([0 n],[0 n/2]);
    gce().children.data(:,3) = max(logC(logC<>%inf))

    // Axes settings
    xgrid(25,1,7)
    ax = gca();
    set(ax, "view","2d", "tight_limits",["on" "on" "off"], "grid_position","foreground");
    xtitle("","","","");
    ax.margins(2) = 0.05;

    // Color bar
    colorbar();
    cb = gce();
    cb.y_ticks.labels = msprintf("$\\Large 10^{%s}$\n", cb.y_ticks.labels);
    title(cb,"$C_n^k$", "fontsize",3)
endfunction

clf
drawlater
// Figure settings
f = gcf();
f.color_map = jet(100);
f.axes_size = [840 570];

// Main plot
ax = drawCnk(0:10:1500); sca(ax);
xtitle("nchoosek(n, k)","n","k","$C_n^k$");
set([ax.title ax.x_label ax.y_label ax.z_label], "font_size",4);
xstring(1250, 450, "Overflow")
gce().font_size = 4;

// Inset
xsetech([0.11 0.11 0.37 0.42]);
ax = drawCnk(0:106);
ax.sub_ticks = [3 3];
gce().sub_ticks(2) = 4;
drawnow
   ]]></scilab:image>
   <para>
        Going beyond, in logarithmic mode :
   </para>
   <programlisting role="example"><![CDATA[
// /!\ : The drawCnk() function used here is defined in the previous example.

clf
drawlater

// Figure settings
f = gcf();
f.color_map = jet(100);
//f.axes_size = [840 570];

// Main plot
ax = drawCnk(0:10000:1e6); sca(ax);
xtitle("nchoosek(n, k)","n","k","$C_n^k$");
set([ax.title ax.x_label ax.y_label ax.z_label], "font_size",4);

// Inset
xsetech([0.12 0.11 0.37 0.42]);
ax = drawCnk(0:1000:1e5);
ax.sub_ticks = [3 3];
gce().sub_ticks(2) = 4;

drawnow
   ]]></programlisting>
   <scilab:image><![CDATA[
function ax= drawCnk(n)
    // Preparing data
    [N,K] = meshgrid(n);
    noOp = K>N;
    K(noOp) = 0;
    [logC, C] = nchoosek(N, K);
    C(noOp) = %nan;
    if max(n)<2000, logC = log10(C), end

    // Surface of Cnk data
    surf(N, K, logC);
    gce().color_mode = -2; // hidding the mesh
    plot2d([0 n],[0 n/2]);
    gce().children.data(:,3) = max(logC(logC<>%inf))

    // Axes settings
    xgrid(25,1,7)
    ax = gca();
    set(ax, "view","2d", "tight_limits",["on" "on" "off"], "grid_position","foreground");
    xtitle("","","","");
    ax.margins(2) = 0.05;

    // Color bar
    colorbar();
    cb = gce();
    cb.y_ticks.labels = msprintf("$\\Large 10^{%s}$\n", cb.y_ticks.labels);
    title(cb,"$C_n^k$", "fontsize",3)
endfunction

clf
drawlater

// Figure settings
f = gcf();
f.color_map = jet(100);
f.axes_size = [840 570];

// Main plot
ax = drawCnk(0:10000:1e6+1e4); sca(ax);
xtitle("nchoosek(n, k)","n","k","$C_n^k$");
set([ax.title ax.x_label ax.y_label ax.z_label], "font_size",4);

// Inset
xsetech([0.12 0.11 0.37 0.42]);
ax = drawCnk(0:1000:1e5);
ax.sub_ticks = [3 3];
gce().sub_ticks(2) = 4;

drawnow
   ]]></scilab:image>
   <para>
        Top line <literal>C<subscript>n</subscript><superscript>n/2</superscript></literal> :
        <literal>nchoosek(n, n/2)</literal> and its known close approximation
        <latex alt="2^n /sqrt(pi.n/2)">2^n / \sqrt{\pi n/2}</latex>
   </para>
   <programlisting role="example"><![CDATA[
n = round(10^(1:0.1:8))*2;
logb = nchoosek(n, n/2, "log10");
clf
plot2d("ll", n, logb)
ax = gca();
xgrid(color("grey70"), 1, 7);
//ax.sub_ticks = [ 8 0];
ax.tight_limits = "on";
ax.y_ticks.labels = msprintf("$\\Large 10^{%d}$\n", ax.y_ticks.locations);
xlabel("n", "fontsize",4);
xstring(60, 1000, "nchoosek(n, n/2)", 270)
set(gce(), "clip_state", "off", "font_size",3);

// Relative difference with the 2^n / sqrt(pi.n/2) approximation:
e = abs(10.^(n*log10(2) - (log10(%pi)+log10(n/2))/2 - logb) - 1);

ax = newaxes();
ax.filled = "off";
ax.y_location = "right";
ax.tight_limits = ["on" "off"];
ax.font_color = color("magenta");
plot2d("ll", n, e, style=color("magenta"))
ax.axes_visible(1) = "off";

legend("$\large \left|{\frac{2^n}{\sqrt{\pi n/2}} / nchoosek(n, n/2)}-1\right|$", ..
        "in_upper_left", %f);
   ]]></programlisting>
   <scilab:image><![CDATA[
        n = round(10^(1:0.1:8))*2;
        logb = nchoosek(n, n/2, "log10");
        clf
        plot2d("ll", n, logb)
        ax = gca();
        xgrid(color("grey70"), 1, 7);
        //ax.sub_ticks = [ 8 0];
        ax.tight_limits = "on";
        ax.y_ticks.labels = msprintf("$\\Large 10^{%d}$\n", ax.y_ticks.locations);
        xlabel("n", "fontsize",4);
        xstring(60, 1000, "nchoosek(n, n/2)", 270)
        set(gce(), "clip_state", "off", "font_size",3);

        // Relative difference with the 2^n / sqrt(pi.n/2) approximation:
        e = abs(10.^(n*log10(2) - (log10(%pi)+log10(n/2))/2 - logb)-1);

        ax = newaxes();
        ax.filled = "off";
        ax.y_location = "right";
        ax.tight_limits = ["on" "off"];
        ax.font_color = color("magenta");
        plot2d("ll", n, e, style=color("magenta"))
        ax.axes_visible(1) = "off";

        legend("$\large \left|{\frac{2^n}{\sqrt{\pi n/2}} / nchoosek(n, n/2)}-1\right|$", ..
                "in_upper_left", %f);
   ]]></scilab:image>
</refsection>
    <refsection>
        <title>Bibliography</title>
            <itemizedlist>
                <listitem>
                    <ulink url="http://en.wikipedia.org/wiki/Binomial_coefficients">Binomial coefficients (Wikipedia)</ulink>
                </listitem>
                <listitem>
                    Boost C++ librairies, Binomial Coefficients, 2006 , 2007, 2008, 2009, 2010 :
                    John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande,
                    Johan Råde, Gautam Sewani and Thijs van den Berg.
                </listitem>
                <listitem>
                    "Introduction to discrete probabilities with Scilab", Michael Baudin, 2010
                </listitem>
            </itemizedlist>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="factorial">factorial</link>
            </member>
            <member>
                <link linkend="gamma">gamma</link>
            </member>
            <member>
                <link linkend="perms">perms</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>History</title>
        <revhistory>
            <revision>
                <revnumber>6.1.0</revnumber>
                <revdescription>
                    <literal>nchoosek()</literal> introduced.
                </revdescription>
            </revision>
        </revhistory>
    </refsection>
</refentry>
