Thread: Summation Notation in Generalized Bernoulli Trials

So I'm reading through Robert Ash's Basic Probability Theory and I see the summation notation reproduced in the picture below.



I'm not familiar with this sort of writing--are we summing first from to some assumed upper bound like , then do it again from and so on?

If so then I'm have a hard time seeing how the first equality is supposed to be true. I would think the RHS would come out p_{1}^{n} + ...

Hello.

The notation means that for each tuple (n_1,n_2,...,n_k) of non-negative integers, such that n_1+n_2+...+n_k=n, we get a term in the sum.

The order of the terms in the sum does not matter, but if it makes it any clearer, here are some examples of what the tuple (n_1,n_2,...,n_k) might be:

(n,0,0,...,0),
(n-1,1,0,0,...,0),
(n-2,1,1,0,0,...,0),
...
(n-2,2,0,0,...0),
...
(0,0,...,0,n).

So for every such tuple, you get a term on the left hand side.