<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>9.0.11 (Binomial coefficients)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h1>Binomial coefficients</h1>

<p>Now that we are familiar with Pascal's triangle, we can proceed to introduce binomial coefficients. There are several different definitions of a binomial coefficient and we will see how we can obtain these definitions via combinatorial arguments. For now we will just say that binomial coefficients are the numbers \({n}\choose{k}\) that make up the entries of Pascal's triangle. These numbers can be calculated by the following formula
\[
nCr = {n\choose r} = \frac{n!}{r!(n-r)!}.
\]</p>
<br/>
<h3>Examples</h3>

<p>Calculate the following binomial coefficients</p>
<ol type="a">
<li>\({12}\choose{4}\)</li>
<li>\({n}\choose{0}\)</li>
<li>\({n}\choose{1}\)</li>
<li>\({n}\choose{n}\)</li>
</ol>

<h3>Solutions</h3>

<ol type="a">
<li>We input the required values into the formula above, so we have
\[
{12}\choose{4}  
\]
\[
=\frac{12!}{4!8!} = \frac{479001600}{24 \times 40320} = \frac{479001600}{967680} = 495.
\]</li>
<li>Here we have 
\[
{n}\choose{0} 
\]
\[
= \frac{n!}{0!(n-0!)} = \frac{n!}{n!} = 1.
\]
So any binomial coefficient of the form \({n}\choose{0}\) evaluates to \(1\)</li>
<li>Here we have 
\[
{n}\choose{1} 
\]
\[
=\frac{n!}{1!(n-1!)} = \frac{n!}{(n-1)!} = \frac{n \times n-1 \times n-2 \times \cdots \times 1}{ n-1 \times n-2 \times \cdots \times 1} = n.
\]
Thus any binomial coefficient of the form \({n}\choose{1}\) evaluates to the top number \(n\).</li>
<li>Here we have 
\[
{n}\choose{n} 
\]
\[
=\frac{n!}{n!(n-n)!} = 0! = 1. 
\]
Thus any binomial coefficient of the form \({n}\choose{n}\) evaluates to \(1\) also.</li>
</ol>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
