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
Using induction and the Binomial Theorem
We have an arbitrary number of variables to the power of . For the sake of simplicity, I will use a small example. The problem could be asking for the number of terms in
. Since all equivalent parts can be combined, we only need to worry about different variables. Those variables must be in terms of
,
, and
. It's easy to see why the exponent value of each variable must sum to
(Imagine k groups. Pick one variable from each group). Our problem then becomes
+
+
=
, where
,
, and
are the exponents of
,
, and
. This is a straightforward application of the Binomial Theorem. Thus, our answer is
. A more generalized form would be
.
Combinatorial proof
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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)
Olympiad
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