Question

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+........(1+x{)}^{30}$ is:

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

The given sequence is geometrical progression with common ratio (1+x)
$(1+x{)}^{21}+(1+x{)}^{22}+.....(1+x{)}^{30}=(1+x{)}^{21}(1+(1+x)+......(1+x{)}^9)$
=$(1+x{)}^{21}(\frac{(1+x{)}^{10}-1}{(1+x)-1})$
=$(1+x{)}^{21}(\frac{(1+x{)}^{10}-1}{x})$
=$\frac{(1+x{)}^{31}-(1+x{)}^{21}}{x})$
Coefficient of $x^5$ in $\frac{(1+x{)}^{31}-(1+x{)}^{21}}{x})$ will be equal to coefficient of $x^6$ in $(1+x{)}^{31}-(1+x{)}^{21}$, which is ${}^{31}{C}_6.{}^{-21}{C}_6$.