The mean of the coefficients of x^n, x^n+1,, x^r in the binomial expansion of (2 + x)^r is:

The mean of the coefficients of x^n, x^n+1,, x^r in the binomial… — Solution | JEE Mathematics

MCQEasyJEE 2023Binomial Expansion

The mean of the coefficients of $x^n, x^{n+1}, \dots, x^r$ in the binomial expansion of $(2 + x)^r$ is:

Correct answer:A

Standard Method

Given: The binomial expansion is $(2 + x)^r$ .

Find: The mean of the coefficients mentioned in the expansion.

From the solution, the expansion is written as

(2 + x)^r = \sum_{k=0}^{r} \binom{r}{k} 2^{r-k} x^k

So the coefficient of $x^n$ is $\binom{r}{n} 2^{r-n}$ , and similarly for the required terms.

The mean is taken as the average of the coefficients, so

\text{Mean} = \frac{\sum_{k=0}^{r} \binom{r}{k} 2^{r-k}}{r+1}

From the extracted solution,

\text{Mean} = \frac{19152}{7} = 2736

Therefore, the mean of the coefficients is $2736$ and the correct option is A.

Taking the mean as the sum of coefficients only is incorrect, because mean requires division by the number of terms. Always divide the total by the appropriate count of terms.
Confusing the coefficient of $x^k$ in $(2+x)^r$ with just $\binom{r}{k}$ is wrong, because the factor $2^{r-k}$ is also present. Use the full coefficient $\binom{r}{k}2^{r-k}$ .

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