Question

Find the coefficient of x^4 in the expansion of (1 + 2x + 3x^3)^10?

Answer 1

$(1+2x+3x^3)^{10}$

The only possible configurations of terms that contribute to generating the $x^4$ term in the expansion are given by the multinomial theorem to be

$\left(\dbinom{10}{6,4,0}(1)^6(2x)^4(3x^3)^0+\dbinom{10}{8,1,1}(1)^8(2x)^1(3x^3)^1\right)x^4$

where

$\dbinom{10}{k_1,\ldots,k_m}=\dfrac{10!}{k_1!\cdots k_m!}$

Simplifying a bit gives a coefficient of

$\dbinom{10}{6,4,0}(2)^4+\dbinom{10}{8,1,1}(2)(3)$
$16\dbinom{10}{6,4,0}+6\dbinom{10}{8,1,1}$
$16\dfrac{10!}{6!4!0!}+6\dfrac{10!}{8!1!1!}$
$16(5\times3\times2\times7)+6(10\times9)=3900$