Question

Consider the expansion of $(1+x{)}^{2n+1}$
The average of the coefficients of the two middle terms in the expansion is

• A. ${}^{2n+1}{C}_{n+2}$
• B. ${}^{2n+1}{C}_n$
• C. ${}^{2n+1}{C}_{n-1}$
• D. ${}^{2n}{C}_{n+1}$

Answer 1

The correct option is B ${}^{2n+1}{C}_n$

Since, $2n+1$ is odd.

Hence, $\frac{2n+1+1}{2}$ and $\frac{2n+1+3}{2}$ are two middle terms.

i.e. $(n+1)th$ and $(n+2)th$ terms are two middle terms.

$\therefore \frac{C_n^{2n+1}+C_{n+1}^{2n+1}}{2}=\frac{C_{n+1}^{2n+1+1}}{2}$

$=\frac{1}{2}{C}_{n+1}^{2n+2}=\frac{1}{2}.\frac{2n+2}{n+1}{C}_n^{2n+1}=C_n^{2n+1}$

Hence, B is the correct option.