Binomial Theorem - Sums of Coefficients in Polynomial Expansions
Binomial Theorem - Sums of Coefficients in Polynomial Expansions
IP 4
Show that \(\left( \begin{matrix}
n \\
0 \\
\end{matrix} \right)+\left( \begin{matrix}
n \\
1 \\
\end{matrix} \right)+\ldots +\left( \begin{matrix}
n \\
n-1 \\
\end{matrix} \right)+\left( \begin{matrix}
n \\
n \\
\end{matrix} \right)={{2}^{n}}\), for integer \(n\ge 1\).
Using the result in (i) and the binomial expansions of \({{\left( 1+1 \right)}^{n}}\) and \({{\left( 1-1 \right)}^{n}}\), or otherwise, show that \(\sum\limits_{r\,\,\mathrm{even}}^{n}{\left( \begin{matrix}
n \\
r \\
\end{matrix} \right)}=\sum\limits_{r\,\,\mathrm{odd}}^{n}{\left( \begin{matrix}
n \\
r \\
\end{matrix} \right)}\), for integer \(n\ge 1\).
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Answer:(i) \(\displaystyle\sum_{r=0}^{n}\binom{n}{r}=2^n\) (ii) \(\displaystyle\sum_{r\text{ even}}^{n}\binom{n}{r}=\sum_{r\text{ odd}}^{n}\binom{n}{r}=2^{n-1}\)