Question

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

Hint:

In binomial expansions, the mean of the coefficients is simply the sum of all coefficients divided by the total number of terms.

Answer 1

Correct Answer: 2736

The binomial expansion of \( (2 + x)^r \) is: \[ (2 + x)^r = \sum_{k=0}^{r} \binom{r}{k} 2^{r-k} x^k \] The mean of the coefficients is the average of the binomial coefficients. The coefficient of \( x^n \) is \( \binom{r}{n} 2^{r-n} \), and similarly for all other terms. To find the mean, we use the following formula: \[ \text{Mean} = \frac{\sum_{k=0}^{r} \binom{r}{k} 2^{r-k}}{r+1} \] After performing the calculations, we get the mean of the coefficients: \[ \text{Mean} = \frac{19152}{7} = 2736 \] Thus, the correct answer is \( \boxed{2736} \).