Difference between revisions of "Multinomial Theorem"
(→Combinatorial proof) |
Twod horse (talk | contribs) m |
||
Line 44: | Line 44: | ||
===Olympiad=== | ===Olympiad=== | ||
− | |||
− | |||
− | |||
[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Combinatorics]] | [[Category:Combinatorics]] | ||
[[Category:Multinomial Theorem]] | [[Category:Multinomial Theorem]] |
Revision as of 02:59, 12 February 2021
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
[hide]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)