Mathematics - Binomial Theorem Question with Solution | TestHub

$\because (1+x{)}^2=1+2x+x^2$, $${\left(1+x^2\right)}^3=1+3x^2+3x^4+x^6,$$ and ${\left(1+x^3\right)}^4=1+4x^3+6x^6+4x^9+x^{12}$ 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 .