Question

The coefficient of $x^{10}$ in the expansion of $(1+x{)}^2(1+x^2{)}^3(1+x^3{)}^4$ is equal to :

• A. $52$
• B. $44$
• C. $50$
• D. $56$

Answer 1

Answer: (A)

$52$

$(1+x{)}^2=1+2x+x^2$

$(1+x^2{)}^3=1+3x^2+3x^4+x^6$

$(1+x^3{)}^4=1+4x^3+6x^6+4x^9+x^{12}$

From the above the binomial expansion, we want the terms containing $x^{10}$ after multiplication

So, the combinations are:

$x.x^9,x.x^6.x^3,x^2.x^2.x^6,x^4.x^6$

Their coefficients are $2\times 4,2\times 1\times 4,1\times 3\times 6,3\times 6$

Which is $8,8,18,18$

Sum of the coefficients is $8+8+18+18=52$

Hence, the coefficient of $x^{10}$ in the expansion of $(1+x{)}^2(1+x^2{)}^3(1+x^3{)}^4$ is $52$