Skolem–Noether theorem: Difference between revisions

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In [[mathematics]], a '''trinomial expansion''' is the expansion of a power of a sum of three terms into [[monomial]]s. The expansion is given by
:<math>(a+b+c)^n = \sum_{i,j,k}  {n \choose i,j,k}\, a^i \, b^j \, c^k </math>
where ''n'' is a nonnegative integer and the sum is taken over all combinations of nonnegative indices ''i'', ''j'', and ''k'' such that ''i''&nbsp;+&nbsp;''j''&nbsp;+&nbsp;''k''&nbsp;=&nbsp;''n''. The '''trinomial coefficients''' are given by
:<math> {n \choose i,j,k} = \frac{n!}{i!\,j!\,k!} \,.</math>
 
This formula is a special case of the [[multinomial formula]] for ''m''&nbsp;=&nbsp;3. The coefficients can be defined with a generalization of [[Pascal's triangle]] to three dimensions, called [[Pascal's pyramid]] or Pascal's tetrahedron.
 
The number of terms of an expanded trinomial is
:<math> \frac{(n+2)(n+1)}{2} </math>
where ''n'' is the exponent to which the trinomial is raised.
 
==See also==
* [[Binomial expansion]]
* [[Pascal's pyramid]]
* [[Multinomial coefficient]]
 
{{DEFAULTSORT:Trinomial Expansion}}
[[Category:Factorial and binomial topics]]

Latest revision as of 12:08, 24 July 2014

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