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}=(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(10!)/(6!4!0!)+6(10!)/(8!1!1!)`

`16(5*3*2*7)+6(10*9)=3900`