The coefficient of x
Question:

The coefficient of $x^{5}$ in the expansion of $(1+x)^{21}+(1+x)^{22}+\ldots+(1+x)^{30}$

(a) 51C5

(b) 9C5

(c) 31C6 − 21C6

(d) 30C5 + 20C5

Solution:

(c) ${ }^{31} C_{6}-{ }^{21} C_{6}$

We have $(1+x)^{21}+(1+x)^{22}+\ldots(1+x)^{30}$

$=(1+x)^{21}\left[\frac{(1+x)^{10}-1}{(1+x)-1}\right]$

$=\frac{1}{x}\left[(1+x)^{31}-(1+x)^{21}\right]$

Coefficient of $x^{5}$ in the given expansion $=$ Coefficient of $x^{5}$ in $\frac{1}{x}\left[(1+x)^{31}-(1+x)^{21}\right]$

$=$ Coefficient of $x^{6}$ in $\left[(1+x)^{31}-(1+x)^{21}\right]$

$={ }^{31} C_{6}-{ }^{21} C_{6}$