Q. The coefficient of $x^{4}$ in the expansion of ${(1 + x + x^{2} + x^{3})}^{n}$ is
A) $\ ^{n}C_{4}$
B) $\ ^{n}C_{4} + \ ^{n}C_{2}$
C) $\ ^{n}C_{4} + \ ^{n}C_{2} + \ ^{n}C_{2}$
D) $\ ^{n}C_{4} + \ ^{n}C_{2} + {\ ^{n}C_{1} \bullet \ }^{n}C_{2}$
Explanation
(d)
$\left( 1 + x + x^{2} + x^{3} \right)^{n} = \left\{ (1 + x)^{n}\left( 1 + x^{2} \right)^{n} \right\}$
$= \left( 1 + \ ^{n}C_{1}x + \ ^{n}C_{2}x^{2} + ... + \ ^{n}C_{n}x^{n} \right)(1 + \ ^{n}C_{1}x^{2} + \ ^{n}C_{2}x^{4} + ... + \ ^{n}C_{n}x^{2n})$
Therefore the coefficient of $x^{4} = \ ^{n}C_{2} + \ ^{n}C_{2}\ ^{n}C_{1} + \ ^{n}C_{4}$
$= \ ^{n}C_{4} + \ ^{n}C_{2} + \ ^{n}C_{1}\ ^{n}C_{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.