The coefficient of $x^{10}$ in the expansion of $(1+x)^2$ $\left(1+x^2\right)^3\left(1+x^3\right)^4$ is…
The coefficient of $x^{10}$ in the expansion of $(1+x)^2$ $\left(1+x^2\right)^3\left(1+x^3\right)^4$ is equal to
52
44
50
56
Solution
$\because(1+x)^2=1+2 x+x^2$,
$
\left(1+x^2\right)^3=1+3 x^2+3 x^4+x^6,
$
and $\left(1+x^3\right)^4=1+4 x^3+6 x^6+4 x^9+x^{12}$
So, the possible combinations for $x^{10}$ are:
$
x \cdot x^9, x \cdot x^6 \cdot x^3, x^2 \cdot x^2 \cdot x^6, x^4 \cdot x^6
$
Corresponding coefficients are $2 \times 4,2 \times 1 \times$ $4,1 \times 3 \times 6,3 \times 6$ or $8,8,18,18$.
$\therefore$ Sum of the coefficient is
$
8+8+18+18=52
$
Therefore, the coefficient of $x^{10}$ in the expansion of $(1+x)^2\left(1+x^2\right)^3\left(1+x^3\right)^4$ is 52