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.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.