Q. The coefficient of $t^{24}$ in the expansion of ${(1 + t}^{2})^{12}\left( 1 + t^{12} \right)(1 + t^{24})$ is
A) $\ ^{12}C_{6}$ $+ 2$
B) $\ ^{12}C_{5}$
C) $\ ^{12}C_{6}\$
D) $\ ^{12}C_{7}\ \$
Explanation
(a)
We have, ${(1 + t}^{2})^{12}{(1 + t}^{12}){(1 + t}^{24})$
$= {\ 1 +}^{12}C_{1}t^{2} + \ ^{12}C_{2}t^{4} + \ ^{12}C_{3}t^{6} +$…$+ \ ^{12}C_{6}t^{12} +$…)$\left( 1 + t^{12} + t^{24} + t^{36} \right)$
$\therefore$ Coefficient of $t^{24}$ in ${(1 + t}^{2})^{12}\left( {1 + t}^{12} \right)\left( 1 + t^{24} \right) = \ ^{12}C_{6} + 2$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.