Question

Co-efficient of $x^5$ in the expansion of $(1+x^2{)}^5(1+x{)}^4$ is ___________.

• A. 40
• B. 50
• C. 60
• D. 30

Answer 1

The correct option is C

60

The first few terms of the expansion of $(1+x^2{)}^5$ are $1+{}^5{C}_1{x}^2+{}^5{C}_2{x}^4.....$ the terms after this has power of x greater than 5. So we will consider only these while calculating the coefficient of $x^5$. From $(1+x{)}^4$ we will consider the terms whose power of x are 5,3 and 1 corresponding to 0,2 and 4 from $1+{}^5{C}_1{x}^2+{}^5{C}_2{x}^4......$ so that they will add up to 5.
Coefficient of $x^5$ in the expansion of $(1+x{)}^4$ is zero.
Coefficient of $x^3$ in the expansion of $(1+x{)}^4$ is ${}^4{C}_3$.
Coefficient of $x^1$ in the expansion of $(1+x{)}^4$ is ${}^4{C}_1$.
We have to multiply the corresponding coefficients and add to get the coefficient of $x^5$.
=$1\times 0+{}^4{C}_3\times {}^5{C}_1+{}^4{C}_1\times {}^5{C}_2$
=$4\times 5+4\times 10$
=60