So I have the following which I must prove :
$$\sum_{(n_1,n_2,n_3)\,:\,n_1+n_2+n_3=n} \binom{n}{n_1, n_2, n_3} = 3^n$$
I'm not sure where I must begin. This is a multinomial.
2025-01-13 05:49:01.1736747341
The sum which gives $3^n$
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Use the multinomial theorem as suggested by Carl Heckman or you can think the equality as two different ways to represent the following combination:
Suppose you have $n$ objects and you want to assign them to three clusters. Both the RHS and LHS are the total number of assignments that you can made.