The mean of the coefficients of $$x, x^2, \ldots, x^7$$ in the binomial expression of $$(2 + x)^9$$ is _______

The binomial expansion of $$(2 + x)^9$$ is:

$$(2+x)^9 = \sum_{r=0}^{9}\binom{9}{r}2^{9-r}x^r$$

We need the mean of the coefficients of $$x, x^2, \ldots, x^7$$.

Sum of all coefficients (put $$x = 1$$): $$(2+1)^9 = 3^9 = 19683$$

Coefficient of $$x^0 = \binom{9}{0}2^9 = 512$$

Coefficient of $$x^8 = \binom{9}{8}2^1 = 18$$

Coefficient of $$x^9 = \binom{9}{9}2^0 = 1$$

Sum of coefficients from $$x^1$$ to $$x^7$$:

$$= 19683 - 512 - 18 - 1 = 19152$$

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