Question

If $n$is even and the value of ${}^n{C}_r$ is maximum, then $r=$

• A. $\frac{n}{2}$
• B. $\frac{n+1}{2}$
• C. $\frac{n-1}{2}$
• D. None of these

Answer 1

The correct option is A

$\frac{n}{2}$

Explanation for correct option:

Maximum value of ${}^n{C}_r$:

Apply the formula for binomial coefficient series,

${}^{n}C_0,{}^{n}C_1,{}^{n}C_2,\dots .,{}^{n}C_r,\dots .,{}^{n}C_{n-2},{}^{n}C_{n-1},{}^{n}C_n$

Here, we have to find the maximum value of value of ${}^n{C}_r$

And maximum value will be the middle binomial coefficient.

Total number of terms$=n+1;r=0$ to $n$

If we compare, ${}^n{C}_r$ is $(r+1)$th term

And middle term$=\frac{(n+1+1)}{2}=\frac{n}{2}+1$

$\Rightarrow r+1=\frac{n}{2}+1$

$\Rightarrow r=\frac{n}{2}$

Hence, option A is correct.