The Multinomial Theorem states that where is the multinomial coefficient .
Note that this is a direct generalization of the Binomial Theorem: when it simplifies to
Proof by Induction
Proving the Multinomial Theorem by Induction
For a positive integer and a non-negative integer ,
When the result is true, and when the result is the binomial theorem. Assume that and that the result is true for When Treating as a single term and using the induction hypothesis: By the Binomial Theorem, this becomes: Since , this can be rewritten as:
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- The expression
is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
(Source: 2006 AMC 12A Problem 24)