Multinomial Theorem
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
Contents
Proof
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:
Combinatorial proof
This article is a stub. Help us out by expanding it.
Problems
Intermediate
- 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)