Difference between revisions of "Multinomial Theorem"
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(a_1 + a_2)^n = \sum_{\substack{0\leq j_1, j_2 \leq n \ j_1 + j_2 = n}} \binom{n}{j_1; j_2} a_1^{j_1}a_2^{j_2} = \sum_{j = 0}^n \binom{n}{j} a_1^j a_2^{n - j} | (a_1 + a_2)^n = \sum_{\substack{0\leq j_1, j_2 \leq n \ j_1 + j_2 = n}} \binom{n}{j_1; j_2} a_1^{j_1}a_2^{j_2} = \sum_{j = 0}^n \binom{n}{j} a_1^j a_2^{n - j} | ||
</cmath> | </cmath> | ||
+ | |||
+ | == Proof == | ||
+ | === Using [[induction]] and the Binomial Theorem === | ||
+ | {{incomplete | section}} | ||
+ | === Combinatorial proof === | ||
+ | {{incomplete | section}} | ||
==Problems== | ==Problems== | ||
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[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Combinatorics]] | [[Category:Combinatorics]] | ||
+ | [[Category:Algebra]] |
Revision as of 10:50, 30 April 2008
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
Using induction and the Binomial Theorem
Combinatorial proof
Problems
Introductory
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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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This article is a stub. Help us out by expanding it.