While tutoring today a student asked me about the following identity

.

One way to prove it is to count the number of possible triples of sets with . This can be done in two ways. First, if contain elements respectively then the number of such triples clearly equals the left-hand side of . On the other hand, since each element of must belong to one of the four disjoint regions in the picture below

the number of such triples equals precisely .

An algebraic proof can be obtained by repeated use of the binomial theorem:

.

Similarly, one obtains the more general identity

.

This also follows from the multinomial theorem since the left-hand side equals

.

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