Difference between revisions of "Multinomial Theorem"

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The multinomial theorem states that
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The '''Multinomial Theorem''' states that
  
<math>(a_1+a_2+\cdots+a_x)^n=\sum_{k_1,k_2,\cdots,k_x}\binom{n}{k_1,k_2,\cdots,k_x}a_1^{k_1}a_2^{k_2}\cdots a_x^{k_x}</math>
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<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
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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}{k_1,k_2,\cdots,k_x}=\dfrac{n!}{k_1!k_2!\cdots k_x!}</math>
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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  
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*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:Number Theory]]
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[[Category:Combinatorics]]

Revision as of 18:02, 29 April 2008

The Multinomial Theorem states that

\[(a_1+a_2+\cdots+a_k)^n=\sum_{\substack{j_1,j_2,\ldots,j_k \\ 0 \leq j_i \leq n \textrm{ for each } i \\ \textrm{and } j_1 + \ldots + j_k = n}}\binom{n}{j_1; j_2; \ldots ; j_k}a_1^{j_1}a_2^{j_2}\cdots a_k^{j_k}\]

where $\binom{n}{j_1; j_2; \ldots ; j_k}$ is the multinomial coefficient $\binom{n}{j_1; j_2; \ldots ; j_k}=\dfrac{n!}{j_1!\cdot j_2!\cdots j_k!}$.

Note that this is a direct generalization of the Binomial Theorem: when $k = 2$ it simplifies to \[(a_1 + a_2)^n = \sum_{\substack{0\leq j_1, j_2 \leq n \\ j_1 + j_2 = n}} \binom{n}{j_1; j_2} a_1^{j_1}a_2^{j_2} = \sum_{j = 0}^n \binom{n}{j} a_1^j a_2^{n - j}.\]

Problems

Introductory

Intermediate

$(x+y+z)^{2006}+(x-y-z)^{2006}$

is simplified by expanding it and combining like terms. How many terms are in the simplified expression?

$\mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qquad \mathrm{(D) \ } 1,008,016\qquad\mathrm{(E) \ }  2,015,028$

-(2006 AMC 12A Problem 24)

Olympiad

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