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<p>Evaluate the following without using a calculator</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\binom{10}{3}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\binom{n+2}{2}$ in terms of $n$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\binom{n+1}{n}$ in terms of $n$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\binom{n}{n}$ in terms of $n$</span></div></li></ol>

Evaluate the following without using a calculator

Binomial Coefficients and Pascal's Triangle

Binomial Theorem

<p>For a positive integer $n$:</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">$(a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r$</p></div></div>
<p>There are $n+1$ terms. Powers of $a$ decrease as powers of $b$ increase; their sum is $n$.</p>
<p class="paragraph-header">Binomial coefficients</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">$\binom nr=\dfrac{n!}{r!(n-r)!}\qquad r=0,1,\ldots,n$</p></div></div>
<p>$0!=1\qquad\binom n0=\binom nn=1\qquad\binom n1=n\qquad\binom nr=\binom n{n-r}$</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 50.0000%;"><col style="width: 50.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Coefficient</strong></th><th style="text-align: center; --cell-justify: center"><strong>Evaluation</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$\binom{10}{2}$</td><td style="text-align: center; --cell-justify: center">$\dfrac{10\times9}{2!}=45$</td></tr><tr><td style="text-align: center; --cell-justify: center">$\binom73$</td><td style="text-align: center; --cell-justify: center">$\dfrac{7\times6\times5}{3!}=35$</td></tr></tbody></table></div></div>
<p>Use the calculator $n\mathrm Cr$ function for larger coefficients.</p>

$(a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r$

There are $n+1$ terms. Powers of $a$ decrease as powers of $b$ increase; their sum is $n$.

$\binom nr=\dfrac{n!}{r!(n-r)!}\qquad r=0,1,\ldots,n$

$0!=1\qquad\binom n0=\binom nn=1\qquad\binom n1=n\qquad\binom nr=\binom n{n-r}$

Use the calculator $n\mathrm Cr$ function for larger coefficients.

Binomial Expansion for Positive Integer Powers

<p>Consider the following number pattern:</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770641322976-jrogjksfkrs.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA2NDEzMjI5NzYtanJvZ2prc2ZrcnMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc4OTIxMiwiZXhwIjoxNzkyNzg5NTEyfQ.ZPIZyiAcCbNmAWqOHtcpuKm7kBqhVOQaNPZVNY7S2C8" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">State a rule of the number pattern above.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence complete the following table:</div><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 18.1818%;"><col style="width: 81.8182%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Row Number</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Sum of the Numbers of the Row, $S_n$</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1   = 1   = 2^0$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$2$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 1   = 2   = 2^1$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 2 + 1   = 4   = 2^2$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 3 + 3 + 1   = 8   = 2^3$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 4 + 6 + 4 + 1   = 16   = 2^4$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$6$</p></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$7$</p></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>...</p></td><td style="text-align: center; --cell-justify: center"><p>...</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$n$</p></td><td style="text-align: center; --cell-justify: center"><p>$1 + (n - 1) + ... + (n - 1) + 1 = $ ________</p></td></tr></tbody></table></div></li></ol>

State a rule of the number pattern above.

General Term of a Sequence

 Number Patterns

<p>Evaluate the following without using a calculator.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\binom{6}{2}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\binom{5}{3}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\binom{n+1}{3}$ in terms of $n$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\binom{n}{1}$ in terms of $n$</span></div></li></ol>

Evaluate the following without using a calculator.

<p>Evaluate the following without using a calculator</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\left( \begin{matrix}    8  \\    3  \\ \end{matrix} \right)$</div></li><li><div class="list-item-paragraph">$\left( \begin{matrix}    5  \\    2  \\ \end{matrix} \right)$</div></li><li><div class="list-item-paragraph">$\left( \begin{matrix}    n+1  \\    4  \\ \end{matrix} \right)$ in terms of $n$</div></li><li><div class="list-item-paragraph">$\left( \begin{matrix}    n  \\    1  \\ \end{matrix} \right)$ in terms of $n$</div></li><li><div class="list-item-paragraph">$\left( \begin{matrix}    n+1  \\    n  \\ \end{matrix} \right)$ in terms of $n$</div></li><li><div class="list-item-paragraph">$\left( \begin{matrix}    n  \\    n  \\ \end{matrix} \right)$ in terms of $n$</div></li><li><div class="list-item-paragraph">$\left( \begin{matrix}    n  \\    0  \\ \end{matrix} \right)$ in terms of $n$</div></li></ol>

$\left( \begin{matrix}    8  \\    3  \\ \end{matrix} \right)$

$\left( \begin{matrix}    5  \\    2  \\ \end{matrix} \right)$

$\left( \begin{matrix}    n+1  \\    4  \\ \end{matrix} \right)$ in terms of $n$

$\left( \begin{matrix}    n  \\    1  \\ \end{matrix} \right)$ in terms of $n$

$\left( \begin{matrix}    n+1  \\    n  \\ \end{matrix} \right)$ in terms of $n$

$\left( \begin{matrix}    n  \\    n  \\ \end{matrix} \right)$ in terms of $n$

$\left( \begin{matrix}    n  \\    0  \\ \end{matrix} \right)$ in terms of $n$

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