Question

The coefficient of $X^6$ in the expansion of is $(1+x{)}^{21}+(1+x{)}^{22}+.......+(1+x{)}^{30}$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

The given sequence is a G.P of 10 terms when common ratio is (1+x)

$\therefore$ S = $(1+x{)}^{21}\frac{\left[\right(1+x{)}^{1}0]-1}{1+x-1}$

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

$\therefore$ The coefficient of $x^6$ in S is coefficient of in the numerator = ${}^{31}{C}_7$ - ${}^{21}{C}_7$ $\Rightarrow$ (c)