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

Question

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

Accepted Answer

$(1+x)^2=1+2x+x^2$; $(1+x^2)^3=1+3x^2+3x^4+x^6$; $(1+x^3)^4=1+4x^3+6x^6+4x^9+x^{12}$

Combinations giving $x^{10}$: $x^1\cdot x^9,\ x^1\cdot x^6\cdot x^3,\ x^2\cdot x^2\cdot x^6,\ x^4\cdot x^6$

Coefficients: $2\times4=8$; $2\times1\times4=8$; $1\times3\times6=18$; $3\times6=18$

$$\text{Total}=8+8+18+18=52$$

Solutions · 1