Prove that the coefficient of x^p in the expansion of (a_0 +a_1 x+a_2 x^2 +a_3 x^3 +...+a_k x^k )^n is Σ((n!)/(n_0 !n_1 !n_2 !n_3 !...n_k !))a_0 ^n_0 a_1 ^n_1 a_2 ^n_2 a_3 ^n_3 ...a_k ^n_k where n_0 , n_1 , n_2 , n_3 , ..., n_k are all non- negative integers subject to the conditions n_0 +n_1 +n_2 +n_3 +...+n_k =n and n_1 +2n_2 +3n_3 +...+kn


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Question Number 22316 by Tinkutara last updated on 15/Oct/17

$Prove that the coefficient of x^{p} in the$ $expansion of {(a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + . . . + a_{k} x^{k})}^{n}$ $is Σ \frac{n!}{n_{0}! n_{1}! n_{2}! n_{3}! . . . n_{k}!} a_{0}^{n_{0}} a_{1}^{n_{1}} a_{2}^{n_{2}} a_{3}^{n_{3}} . . . a_{k}^{n_{k}}$ $where n_{0}, n_{1}, n_{2}, n_{3}, . . ., n_{k} are all non -$ $negative integers subject to the$ $conditions n_{0} + n_{1} + n_{2} + n_{3} + . . . + n_{k} = n$ $and n_{1} + 2 n_{2} + 3 n_{3} + . . . + {k n}_{k} = p .$
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