Multinomial Coefficient

Let
The number

of partitions of into sets of sizes is called the multinomial.

Theorem

There are

-partitions of ,
i.e. integer -tuples s.t.

Theorem

Proof

Count the set of permutations of in two ways.
Firstly

Define equivalence relation on
by
if and only if
permute each other on consecutive blocks of size
Thus find:

and each equivalence class has size