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    <name>
      <text>9.1.17 (Proof of binomial coefficient formula)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<p>Now that we have made the formal definition above, we are in a position to prove the formula for calculating a binomial coefficient. The argument, is essentially, an application of more clever counting.</p>
<br/>
<h3>Proposition</h3>

<p>We can evaluate the binomial coefficient 'n choose r' as
\[
\frac{n!}{r!(n-r)!} = {n\choose r}
\]</p>

<h3>Solution</h3>

<p>Let \(S\) be the set \(S = \{1,2,3,\dots, n\}\). Let us count the possible arrangements of \(S\) in a clever manner.</p>
<ol>
<li>Choose a subset of \(S\) with \(r\) elements call it T. By definition there are \({n}\choose{r}\) choices for T.</li>
<li>Choose an arrangement of \(T\), by above, the number of arrangements is \(r!\).</li>
<li>Choose an arrangement of the remaining elements of S. There are \(n-r\) remaining elements so there are \((n-r)!\) different arrangements.</li>
<li>Note that the number of arrangements of \(S\) is \(n!\).</li>
</ol>
<p>The total number of arrangements of \(S\) has to be equal to the product of all the arrangements of the subsets we have chosen above hence
\[
n! = r! \times (n-r)! \times {n\choose r}.
\]
Rearranging this equation we get that <
\[
\frac{n!}{r!(n-r)!} = {n \choose r},
\]
and so the proof is complete.
</p>]]></text>
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