Find the coefficients of x16 in x x2 x3 x4 x5x2 x3 x4

Find the coefficients of x^16 in (x + x^2 + x^3 + x^4 + x^5)(x^2 + x^3 + x^4 + . . .)^5

Solution

let f(x) = (x^2 + x^3 + x^4 + ...)^5

coefficient of x^11 in f(x) = 5!/(4! * 1!)

= 5

coefficient of x^12 in f(x) = 5!/(4! * 1!) + 5!/(3!*2!)

= 15

coefficient of x^13 in f(x) = 35

coefficient of x^14 in f(x) = 70

coefficient of x^15 in f(x) = 126

therefore, answer = 5 + 15 + 35 + 70 + 121 = 251