Question

Coefficient of $x^5$ in the expansion of
$(1+x{)}^{10}.(1+\frac{1}{x}{)}^{20}$ is

• A. ${}^{30}{C}_5$
• B. ${}^{10}{C}_5$
• C. ${}^{20}{C}_5$
• D. ${}^{30}{C}_{20}$

Answer 1

Answer: (A)

${}^{30}{C}_5$

Given that: $(1+x{)}^{10}.(1+\frac{1}{x}{)}^{20}$

which can also be written as:
$(1+x{)}^{10}.(1+\frac{1}{x}{)}^{20}=\frac{(1+x{)}^{30}}{x^{20}}$

Coefficient of $x^5$ in $(1+x{)}^{10}.(1+\frac{1}{x}{)}^{20}$ will be equivalent to finding coefficient of $x^{20+5}$ i.e. $x^{25}$ in $(1+x{)}^{30}$

Using that the coefficient of $x^r$ in $(1+x{)}^n$ is given by ${}^n{C}_r$, we get the final answer to be ${}^{30}{C}_5$