wikipedia proof is a mess and it's not even complete because the base of induction is assumed to be m = 1, which is incorrect! the base of induction is actually m = 2, because

in reducing the case m + 1 to m we use "binomial theorem". (of course, for completeness we can mention the trivial case m = 1, but that cannot be considered as the base!)

before getting into the proof, we first need a very simple fact:

suppose and then:

Proof:by definition:

note that since we have:

now we can prove the multinomial theorem by induction over m assuming that we know binomial theorem. suppose and multinomial theorem is true for m. for m + 1 we have:

now in the above put then since we'll have also since we'll have finally

becomes which is equivalent to because therefore the above sum is equal to: