The coefficient of x^7 in the expression (1+x)^10+x(1+x)^9+x^2(1+x)^8++x^10 is:

Question

The coefficient of $x^7$ in the expression $(1+x)^{10}+x(1+x)^9+x^2(1+x)^8+\cdots+x^{10}$ is:

Accepted Answer

The series is a G.P. with first term $(1+x)^{10}$ and ratio $\dfrac{x}{1+x}$, 11 terms:

$$\sum = (1+x)^{10}\cdot\frac{1-\left(\frac{x}{1+x}\right)^{11}}{1-\frac{x}{1+x}} = (1+x)^{11}-x^{11}.$$

Coefficient of $x^7$: $\binom{11}{7} = \binom{11}{4} = 330$.

Solutions · 1