We will approach this problem using combinatorics. Note that a general term of the expansion would be of the form (without the coefficient)
where the various powers must always sum to (why?).
Now, to evaluate the coefficient of the term in , we consider the multinomial expression in expanded form:
To generate the term in , we must get from terms, from terms and so on. Let us find the number of ways in which this can be done.
First select those multinomials that will contribute : this can be done in ways. Now, from the remaining multinomials, select those multinomials that will contribute : this can be done in ways. Continuing this process, we see that the number of ways to get from from and so on, that is, the number of times the term in will be generated in the expansion is
This is what is known as the general multinomial coefficient. The multinomial expansion can now be written compactly as
where the summation is carried out over all possible combinations of the ‘s such that . For example, in , let us consider some terms in the expansion:
Find the coefficient of in the expansion of .
From the previous example, the general term in the expansion will be
where must be .
Now, is generated whenever The possible values of the triplet can now simply be listed out: