Difference between revisions of "Multinomial Theorem"
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| − | The | + | The '''Multinomial Theorem''' states that |
| − | < | + | <cmath>(a_1+a_2+\cdots+a_k)^n=\sum_{\substack{j_1,j_2,\ldots,j_k \\ 0 \leq j_i \leq n \textrm{ for each } i \\ |
| + | \textrm{and } j_1 + \ldots + j_k = n}}\binom{n}{j_1; j_2; \ldots ; j_k}a_1^{j_1}a_2^{j_2}\cdots a_k^{j_k}</cmath> | ||
| − | where | + | where <math>\binom{n}{j_1; j_2; \ldots ; j_k}</math> is the [[multinomial coefficient]] <math>\binom{n}{j_1; j_2; \ldots ; j_k}=\dfrac{n!}{j_1!\cdot j_2!\cdots j_k!}</math>. |
| − | <math>\binom{n}{ | + | Note that this is a direct generalization of the [[Binomial Theorem]]: when <math>k = 2</math> it simplifies to |
| + | <cmath>(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> | ||
==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
===Intermediate=== | ===Intermediate=== | ||
| − | *The expression | + | *The [[expression]] |
<math>(x+y+z)^{2006}+(x-y-z)^{2006}</math> | <math>(x+y+z)^{2006}+(x-y-z)^{2006}</math> | ||
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{{stub}} | {{stub}} | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
| − | [[Category: | + | [[Category:Combinatorics]] |
Revision as of 18:02, 29 April 2008
The Multinomial Theorem states that
![\[(a_1+a_2+\cdots+a_k)^n=\sum_{\substack{j_1,j_2,\ldots,j_k \\ 0 \leq j_i \leq n \textrm{ for each } i \\ \textrm{and } j_1 + \ldots + j_k = n}}\binom{n}{j_1; j_2; \ldots ; j_k}a_1^{j_1}a_2^{j_2}\cdots a_k^{j_k}\]](http://latex.artofproblemsolving.com/4/5/c/45cb6228eda6d98ac694f0fa5bf19c95eb0edff4.png)
where 
is the multinomial coefficient 
.
Note that this is a direct generalization of the Binomial Theorem: when 
it simplifies to
![\[(a_1 + a_2)^n = \sum_{\substack{0\leq j_1, j_2 \leq n \\ j_1 + j_2 = n}} \binom{n}{j_1; j_2} a_1^{j_1}a_2^{j_2} = \sum_{j = 0}^n \binom{n}{j} a_1^j a_2^{n - j}.\]](http://latex.artofproblemsolving.com/e/5/1/e51cd32ba94721e2f5e36d5daddafab05c22409d.png)
Problems
Introductory
Intermediate
- The expression
is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
Olympiad
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