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<p>Without expanding completely, prove that&nbsp;$ \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0$ for every positive integer $n$.</p>

Without expanding completely, prove that $ \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0$ for every positive integer $n$.

Sums of Coefficients in Polynomial Expansions

Binomial Theorem

<p>In the expansion of ${{\left( \sqrt[3]{{{x}^{2}}}+3{{x}^{2}} \right)}^{n}}$, the sum of coefficients of each term is $992$ more than the sum of the binomial coefficients, $\left( \begin{matrix}
   n  \\
   r  \\
\end{matrix} \right)$, where&nbsp;$r=0,\,\,1,\,\,2,\,\,\ldots ,\,\,n$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value of $n$</div></li><li><div class="list-item-paragraph">Hence find the term with the greatest coefficient in the expansion of ${{\left( \sqrt[3]{{{x}^{2}}}+3{{x}^{2}} \right)}^{n}}$.</div></li></ol>

In the expansion of ${{\left( \sqrt[3]{{{x}^{2}}}+3{{x}^{2}} \right)}^{n}}$, the sum of coefficients of each term is $992$ more than the sum of the binomial coefficients, $\left( \begin{matrix}
   n  \\
   r  \\
\end{matrix} \right)$, where $r=0,\,\,1,\,\,2,\,\,\ldots ,\,\,n$.

Finding Parameters in Binomial Expansions

Greatest Coefficients in Binomial Expansions

<p>It is given that&nbsp;${{x}^{5}}{{\left( x+3 \right)}^{3}}={{a}_{8}}{{\left( x+1 \right)}^{8}}+{{a}_{7}}{{\left( x+1 \right)}^{7}}+\cdots +{{a}_{1}}\left( x+1 \right)+{{a}_{0}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">By differentiating both side with respect to $x$, show that<br><br></div><div class="list-item-paragraph" style="text-align: center">${{x}^{4}}{{\left( x+3 \right)}^{2}}\left( 8x+15 \right)=8{{a}_{8}}{{\left( x+1 \right)}^{7}}+7{{a}_{7}}{{\left( x+1 \right)}^{6}}+\cdots +{{a}_{1}}$.</div></li><li><div class="list-item-paragraph">Hence, or otherwise, find the value of $7{{a}_{7}}+5{{a}_{5}}+3{{a}_{3}}+{{a}_{1}}$.</div></li></ol>

It is given that ${{x}^{5}}{{\left( x+3 \right)}^{3}}={{a}_{8}}{{\left( x+1 \right)}^{8}}+{{a}_{7}}{{\left( x+1 \right)}^{7}}+\cdots +{{a}_{1}}\left( x+1 \right)+{{a}_{0}}$.

<p>Given that&nbsp;${\left(2x + 1\right)^5 = a_0 + a_1x + a_2x^2 + \cdots + a_5x^5}$, find the value of ${\left(a_0 + a_2 + a_4\right)^2 - \left(a_1 + a_3 + a_5\right)^2}$.</p>

Given that ${\left(2x + 1\right)^5 = a_0 + a_1x + a_2x^2 + \cdots + a_5x^5}$, find the value of ${\left(a_0 + a_2 + a_4\right)^2 - \left(a_1 + a_3 + a_5\right)^2}$.

<p>Given that&nbsp;${{\left( 5x-2 \right)}^{7}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a}_{7}}{{x}^{7}}$, find the value of $\left| {{a}_{0}} \right|+\left| {{a}_{1}} \right|+\left| {{a}_{2}} \right|+\cdots +\left| {{a}_{7}} \right|$.</p>

Given that ${{\left( 5x-2 \right)}^{7}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a}_{7}}{{x}^{7}}$, find the value of $\left| {{a}_{0}} \right|+\left| {{a}_{1}} \right|+\left| {{a}_{2}} \right|+\cdots +\left| {{a}_{7}} \right|$.

<p>Given that&nbsp;$\left( 1+2x \right){{\left( 1-x+{{x}^{2}} \right)}^{10}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a}_{21}}{{x}^{21}}$, find the value of ${{a}_{2}}+{{a}_{4}}+\cdots +{{a}_{18}}+{{a}_{20}}$.</p>

Given that $\left( 1+2x \right){{\left( 1-x+{{x}^{2}} \right)}^{10}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a}_{21}}{{x}^{21}}$, find the value of ${{a}_{2}}+{{a}_{4}}+\cdots +{{a}_{18}}+{{a}_{20}}$.

<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$\left( \begin{matrix}
   n  \\
   0  \\
\end{matrix} \right)+\left( \begin{matrix}
   n  \\
   1  \\
\end{matrix} \right)+\ldots +\left( \begin{matrix}
   n  \\
   n-1  \\
\end{matrix} \right)+\left( \begin{matrix}
   n  \\
   n  \\
\end{matrix} \right)={{2}^{n}}$, for integer $n\ge 1$.<br></div></li><li><div class="list-item-paragraph">Using the result in <strong>(i)</strong> and the binomial expansions of ${{\left( 1+1 \right)}^{n}}$ and ${{\left( 1-1 \right)}^{n}}$, or otherwise, show that&nbsp;$\sum\limits_{r\,\,\mathrm{even}}^{n}{\left( \begin{matrix}
   n  \\
   r  \\
\end{matrix} \right)}=\sum\limits_{r\,\,\mathrm{odd}}^{n}{\left( \begin{matrix}
   n  \\
   r  \\
\end{matrix} \right)}$, for integer $n\ge 1$.<br></div></li></ol>

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