JEE Main 2023 (Online) 11th April Morning ShiftQuestion 1
The mean of the coefficients of $x,x^{2},\ldots,x^{7}$ in the binomial expansion of $(2+x)^{9}$ is $\_\_\_\_\_\_$ .
Binomial Theorem · Properties and Sums of Binomial Coefficients
Answer$2736$
Solution
Concept. All coefficients add up to $3^9$; subtract the three that are not needed. Formulas used. (1) Coefficient of $x^k$ in $(2+x)^9$: ${}^{9}C_k\,2^{9-k}$; sum of all of them $=(2+1)^9$Step 1 - Total. $3^9=19683$. Step 2 - Remove $k=0,8,9$. $2^9=512$, ${}^{9}C_8\cdot2=18$, ${}^{9}C_9=1$: $$\sum_{k=1}^{7}{}^{9}C_k\,2^{9-k}=19683-512-18-1=19152.$$ Step 3 - Mean. $\dfrac{19152}{7}=2736$. Answer: 2736