If in the expansion of

Question:

If in the expansion of $(1+x)^{n}$, the coefficients of pth and qth terms are equal, prove that $p+q=n+2$, where $p \neq q$.

Solution:

Coefficients of the $p$ th and $q$ th terms are ${ }^{n} C_{p-1}$ and ${ }^{n} C_{q-1}$ respectively.

Thus, we have :

${ }^{n} C_{p-1}={ }^{n} C_{q-1}$

$\Rightarrow p-1=q-1$ or, $p-1+q-1=n \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{s} \Rightarrow r=s\right.$ or, $\left.r+s=n\right]$

$\Rightarrow p=q$ or, $p+q=n+2$

If $p \neq q$, then $p+q=n+2$

Hence proved.

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