Is there an expansion for the following summation? $$ (a_1 + a_2 + \cdots + a_k)^n $$
Expansion of $ (a_1 + a_2 + \cdots + a_k)^n $
7
$\begingroup$
binomial-coefficients
2 Answers
10
http://en.wikipedia.org/wiki/Multinomial_theorem
This is what you seek.
-
7It's a shame that the binomial theorem is almost never stated so that it is clearly a special case of the multinomial theorem: $$ (x+y)^n = \sum_{p+q=n} \frac{n!}{p!q!} x^p y^q$$ – 2010-10-20
-
0I think if it's just a hyperlink then it should be a comment instead. – 2010-10-20
-
5@muad: Not necessarily. This link perfectly answers the question at hand. See [this](http://meta.math.stackexchange.com/questions/24/answers-that-simply-link-to-a-paper-with-little-or-no-content-in-the-answer-itse) meta thread. – 2010-10-20
0
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.
-
0But the resulting coefficient will be really big. – 2010-10-20
-
2@M.S: Yes, i agree! – 2010-10-20