Question

Coefficient of $t^{24}in(1+t^2{)}^{12}(1+t^{12}\left)\right(1+t^{24})$ is ?

• A. ${}^{12}{C}_6+3$
• B. ${}^{12}{C}_6+1$
• C. ${}^{12}{C}_6$
• D. ${}^{12}{C}_6+2$

Answer 1

The correct option is D ${}^{12}{C}_6+2$

Here, Coefficient of $t^{24}in\left\{\right(1+t^2{)}^{12}(1+t^{12})(1+t^{24})\}$
Coefficient of $t^{24}in\left\{\right(1+t^2{)}^{12}.(1+t^{12}+t^{24}+t^{36})\}$
Coefficient of $t^{24}in\left\{\right(1+t^2{)}^{12}+t^{12}(1+t^2{)}^{12}+t^{24}(1+t^2{)}^{12}\}$
[Negleting $t^{36}(1+t^2{)}^{12}]$
Coefficient of $t^{24}={(}^{12}{C}_{12}{+}^{12}{C}_6{+}^{12}{C}_0)=2{+}^{12}{C}_6$