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. $56$
• C. $50$
• D. $44$

Answer 1

The correct option is A $52$

The coefficient of $x¹⁰$ in the expansion of $(1+x)²(1+x²)³(1+x³)⁴$ is equal to

the coefficient of $x¹⁰$ in the expansion of $(1+x)²(1+x²)³(1+4x³+6x⁶+4x⁹)$

We can ignore the last term in the expansion $(1+x³)⁴$, since its exponent is

greater than 10.

= Coefficient of $x¹⁰$ in the expansion of $(1+x)²(1+x²)³$

$+4\ast$Coefficient of $x⁷$ in the expansion of $(1+x)²(1+x²)³$

$+6\ast$Coefficient of $x⁴$ in the expansion of $(1+x)²(1+x²)³$

$+4\ast$Coefficient of $x$ in the expansion of $(1+x)²(1+x²)³,$

Coefficient of $x¹⁰$ in the expansion of $(1+x)²(1+x²)³=0,$ since the highest degree term in the expansion is $8.$

Coefficient of $x⁷$ in the expansion of $(1+x)²(1+x²)³=$

Coefficient of $x⁷$ in the expansion of $(1+2\ast x+x²)(1+x²)³$

$=2\ast$Coefficient of $x⁶$ in the expansion of $(1+x²)³$

$=2\ast 1=2,$

Coefficient of $x⁴$ in the expansion of $(1+x)²(1+x²)³=$

Coefficient of $x⁴$ in the expansion of $(1+2\ast x+x²)(1+x²)³$

$=1$*Coefficient of $x⁴$ in the expansion of $(1+x²)³ +

1*Coefficient of $x$ in the expansion of $(1+x²)³$

$=3+3=6$

Coefficient of $x$ in the expansion of $(1+2\ast x+x²)(1+x²)³$

$=2\ast$ constant in the expansion of $(1+x²)³$

$=2,$

Thus ,the coefficient of $x¹⁰$ in the expansion of $(1+x)²(1+x²)³(1+x³)⁴$

$=0+4\ast 2+6\ast 6+4\ast 2$

$=52.$

Hence, the option $A$ is the correct answer.