Question

The coefficient of $x^{24}$ in ${(1+x^2)}^{12}\cdot (1+x^{12})\cdot (1+x^{24})$ 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 ${}^{12}{C}_6+2$

Given, ${(1+x^2)}^{12}\cdot (1+x^{12})\cdot (1+x^{24})$
$=(1{+}^{12}{C}_1{x}^2{+}^{12}{C}_2{x}^4{+}^{12}{C}_3{x}^6{+}^{12}{C}_4{x}^8{+}^{12}{C}_5{x}^{10}{+}^{12}{C}_6{x}^{12}.........{+}^{12}{C}_{12}{x}^{24})(1+x^{12}+x^{24}+x^{36})$
Hence coefficient of $x^{24}$ in above expression is,
${=}^{12}{C}_{12}{+}^{12}{C}_6+1{=}^{12}{C}_6+2$