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Is there an expansion for the following summation? $$ (a_1 + a_2 + \cdots + a_k)^n $$

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2 Answers 2

up vote 10 down vote accepted

http://en.wikipedia.org/wiki/Multinomial_theorem

This is what you seek.

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7  
It's a shame that the binomial theorem is almost never stated so that it is clearly a special case of the multinomial theorem: $$ (x+y)^n = \sum_{p+q=n} \frac{n!}{p!q!} x^p y^q$$ –  lhf Oct 20 '10 at 11:02
    
I think if it's just a hyperlink then it should be a comment instead. –  anon Oct 20 '10 at 13:23
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@muad: Not necessarily. This link perfectly answers the question at hand. See this meta thread. –  jericson Oct 20 '10 at 16:23

Since you know what $(a+b)^{n}$ is you can take $A = a_{1} + a_{2} + \cdots + a_{k-1}$ and $B=a_{k}$ and try to simplify the big expression by the binomial theorem.

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But the resulting coefficient will be really big. –  user2468 Oct 20 '10 at 7:20
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@M.S: Yes, i agree! –  anonymous Oct 20 '10 at 7:34

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