Question

If the coefficient of $5^{\text{th}}$ term is numerically the greatest coefficient in the expansion of $(1-x{)}^n,$ then the positive integral value of $n$ is

• A. $9$
• B. $8$
• C. $7$
• D. $10$

Answer 1

The correct option is B $8$

Coefficient of middle term is numerically the greatest coefficient in the expansion of $(1-x{)}^n$ or $(1+x{)}^n.$
${}^n{C}_r$ is maximum at $r=\{\begin{array}{cc}\frac{n}{2},& \text{if}n\text{is even}\\ \frac{n-1}{2}\text{or}\frac{n+1}{2},& \text{if}n\text{is odd}\end{array}$

Given that coefficient of $5^{\text{th}}$ term is the greatest, therefore $5^{\text{th}}$ term is the middle term.
So, there are total $9$ terms.
Hence, $n=8$