Question

If ${\left(1+x+x^2+x^3\right)}^5=a_0+a_{1}x+a_2{x}^2+\dots \dots +a_{15}{x}^{15}$, then $a_{10}$ is equal to

• A. 99
• B. 100
• C. 101
• D. 110

Answer 1

Answer: (C)

Coefficient of $x^{10}\text{ in }{\left(1+x+x^2+x^3\right)}^5$
$a_{10}={\left(1+x+x^2(1+x)\right)}^5=(1+x{)}^5{\left(1+x^2\right)}^5$
$=\left({}^5{C}_0+{}^5{C}_{1}x+{}^5{C}_2{x}^2+{}^5{C}_3{x}^3+{}^5{C}_4{x}^4+{}^5{C}_5{x}^5\right)\left({}^5{C}_0+{}^5{C}_1{x}^2+{}^5{C}_2{x}^4+{}^5{C}_3{x}^6+{}^5{C}_4{x}^8+{}^5{C}_5{x}^{10}\right)$
$={}^5{C}_0{}^5{C}_5+{}^5{C}_2{}^5{C}_4+{}^5{C}_4{}^5{C}_3=1+10\cdot 5+5\cdot 10$
$a_{10}=101$