Animated Solution for Mathematics - Binomial Theorem: The mean of the coefficients of $x,x^2,\dots ,x^7$ in the binomial expression of $(2+x{)}^9$ is —————————

Every binomial expansion has a heartbeat, which we call the general term. For any expression $(a+b{)}^n$, the general term is given by:

In our specific case, where we have $(2+x{)}^9$, the general term becomes $T_{r+1}=\left(\genfrac{}{}{0ex}{}{9}{r}\right)2^{9-r}{x}^r$. This means the coefficient of $x^r$, which we can denote as $C_r$, is simply $\left(\genfrac{}{}{0ex}{}{9}{r}\right)2^{9-r}$.

Most students lose time by calculating $C_1,C_2,\dots ,C_7$ individually. This is a trap; instead, we use the 'Sum of Coefficients' trick. If you have a polynomial $P(x)=a_n{x}^n+\cdots +a_0$, the sum of all coefficients is simply $P(1)$.

By substituting $x=1$ into our original expression $(2+x{)}^9$, we get:

This value represents the sum of every single coefficient from $C_0$ to $C_9$. It is the grand total.

We have the grand total, but we only want the sum of coefficients from $x^1$ to $x^7$. This means our grand total contains three 'unwanted' guests: $C_0$, $C_8$, and $C_9$. We must surgically remove them.

Let us calculate them:

For $C_0$ (where $r=0$):

For $C_8$ (where $r=8$):

For $C_9$ (where $r=9$):

The sum of these unwanted terms is $512+18+1=531$.

Now, we subtract the unwanted sum from the grand total:

This is the sum of the coefficients from $x^1$ to $x^7$. Finally, the question asks for the mean. Since there are $7$ terms (from $1$ to $7$), we divide our sum by $7$:

And there you have it! By using the properties of polynomials rather than brute force, we have arrived at the answer with elegance and precision. The final answer is 2736.