Q. The coefficient of $\lambda^{n}\mu^{n}$ in the expansion of $\left\lbrack (1 + \lambda)(1 + \mu)(\lambda + \mu) \right\rbrack^{n}$ is
A) $\sum_{r = 0}^{n}C_{r}^{2}$
B) $\sum_{r = 0}^{n}C_{r + 2}^{2}$
C) $\sum_{r = 0}^{n}C_{r + 3}^{2}$
D) $\sum_{r = 0}^{n}C_{r}^{3}$
Explanation
(d)
Coefficient of $\lambda^{n}\mu^{n}$ in ${(1 + \lambda)}^{n}{(1 + \mu)}^{n}{(\lambda + \mu)}^{n}$
$=$coefficient of $\lambda^{n}\mu^{n}$in
$\sum_{r = 0}^{n}{\ ^{n}C_{r}\lambda^{r}\sum_{s = 0}^{n}{\ ^{n}C_{s}\mu^{s}}\sum_{t = 0}^{n}{\ ^{n}C_{t}\lambda^{n - t}\mu^{t}}}$
$= {(\ ^{n}C_{0})}^{3} + {(\ ^{n}C_{1})}^{3} + {(\ ^{n}C_{2})}^{3} + ... = \sum_{r = 0}^{n}{(\ ^{n}C_{r})}^{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.