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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} + \dots + a_{k} x^{k})}^{n}

is Σ \frac{n!}{n_{0}! n_{1}! n_{2}! n_{3}! \dots n_{k}!} a_{0}^{n_{0}} a_{1}^{n_{1}} a_{2}^{n_{2}} a_{3}^{n_{3}} \dots a_{k}^{n_{k}}

where n_{0}, n_{1}, n_{2}, n_{3}, \dots, n_{k} are all non -

negative integers subject to the

conditions n_{0} + n_{1} + n_{2} + n_{3} + \dots + n_{k} = n

and n_{1} + 2 n_{2} + 3 n_{3} + \dots + {k n}_{k} = p .