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  4. Which of the following is the greatest?

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Q. Which of the following is the greatest?

Binomial Theorem

Solution:

We know that ${ }^{n} C_{0}^{2}+{ }^{n} C_{1}{ }^{2}+\cdots+{ }^{n} C_{n}{ }^{2}={ }^{2 n} C_{n}$
and ${ }^{n} C_{0}^{2}-{ }^{n} C_{1}^{2}+\cdots+{ }^{n} C_{n}^{2}= =
\begin{cases}
0, & \text{if $n$ is odd} \\[2ex]
{ }^{n} C_{n / 2}(-1)^{n / 2} & \text{if $n$ is even}
\end{cases}$

From these $,{ }^{31} C_{0}^{2}-{ }^{31} C_{1}^{2}+{ }^{31} C_{2}^{2}-\cdots-{ }^{31} C_{31}{ }^{2}=0$
${ }^{32} C_{0}^{2}-{ }^{32} C_{1}^{2}+{ }^{32} C_{2}^{2}-\cdots+{ }^{32} C_{32}^{2}={ }^{32} C_{16}$
${ }^{34} C_{0}^{2}-{ }^{34} C_{1}^{2}+{ }^{34} C_{2}^{2}-\cdots+{ }^{34} C_{32}^{2}=-{ }^{34} C_{17}$
${ }^{32} C_{0}^{2}+{ }^{32} C_{1}^{2}+{ }^{32} C_{2}^{2}-\cdots+{ }^{32} C_{32}{ }^{2}={ }^{64} C_{32}$
Obviously $^{64} C_{32}$, is greatest.