Question

In the expansion of ${(1+x)}^{-3}$, $\left|x\right|<1$, the coefficient of $x^7$ is

• A. -$36$
• B. ${}^8{C}_7$
• C. $45$
• D. none of these

Answer 1

The correct option is A -$36$

For $|x|<1$
$(1+x{)}^{-3}$
$=1-3x+6x^2-\frac{\mathrm{3.4.5}}{3!}{x}^3...(-1{)}^n{x}^n\frac{\mathrm{3.4.5...}(n+2)}{n!}...\infty$
Hence for coefficient of $x^7$, where $n=7$
We get $-\frac{\mathrm{3.4.5...9}}{7!}$
$=-\frac{9!}{2(7!)}$
$=-\frac{8\left(9\right)}{2}$
$=-36$