Question

If $r,k,p\in W$, then $\sum _{r+k+p=10}{}^{30}{C}_r{\cdot }^{20}{C}_k{\cdot }^{10}{C}_p$ is

• A. ${}^{60}{C}_{50}$
• B. ${}^{60}{C}_{30}$
• C. ${}^{60}{C}_{20}$
• D. $\left({}^{30}{C}_{10}\right)\left({}^{30}{C}_{20}\right)$

Answer 1

The correct option is A ${}^{60}{C}_{50}$

The required sum is cofficient of $x^{10}$ in the expansion of $(1+x{)}^{30}\cdot (1+x{)}^{20}\cdot (1+x{)}^{10}=(1+x{)}^{60}$
Hence the cofficient will be
${}^{60}{C}_{10}$