Question

If the coefficient of pth term in the expansion of ${(1+x)}^n$ is p and the coefficient of $(p+1)$th term is q then n =

• A. p + q + 1
• B. p + q – 1
• C. p – q – 1
• D. p – q – 1

Answer 1

The correct option is B p + q – 1

Coefficient of pth term = ${}^n{C}_{P-1}$
Given ${}^n{C}_{P-1}=p\to \left(1\right){,}^n{C}_P=q\to \left(2\right)$
$\frac{\left(1\right)}{\left(2\right)}\Rightarrow P\left({}^n{C}_P\right)=Q\left({}^n{C}_{P-1}\right)\Rightarrow \frac{1}{(n-p)!p!}=q\frac{1}{(n-(p-1)!(p-1)}\Rightarrow \frac{p}{p}=\frac{q}{n-(p-1)}\Rightarrow \frac{q}{n-p+1}$
$\Rightarrow$n = p + q – 1