Trinomial expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by
- [math]\displaystyle{ (a+b+c)^n = \sum_{{i,j,k}\atop{i+j+k=n}} {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 + j + k = n.[1] The trinomial coefficients are given by
- [math]\displaystyle{ {n \choose i,j,k} = \frac{n!}{i!\,j!\,k!} \,. }[/math]
This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.[2]
Derivation
The trinomial expansion can be calculated by applying the binomial expansion twice, setting [math]\displaystyle{ d = b+c }[/math], which leads to
- [math]\displaystyle{ \begin{align} (a+b+c)^n &= (a+d)^n = \sum_{r=0}^{n} {n \choose r}\, a^{n-r}\, d^{r} \\ &= \sum_{r=0}^{n} {n \choose r}\, a^{n-r}\, (b+c)^{r} \\ &= \sum_{r=0}^{n} {n \choose r}\, a^{n-r}\, \sum_{s=0}^{r} {r \choose s}\, b^{r-s}\,c^{s}. \end{align} }[/math]
Above, the resulting [math]\displaystyle{ (b+c)^{r} }[/math] in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index [math]\displaystyle{ s }[/math].
The product of the two binomial coefficients is simplified by shortening [math]\displaystyle{ r! }[/math],
- [math]\displaystyle{ {n \choose r}\,{r \choose s} = \frac{n!}{r!(n-r)!} \frac{r!}{s!(r-s)!} = \frac{n!}{(n-r)!(r-s)!s!}, }[/math]
and comparing the index combinations here with the ones in the exponents, they can be relabelled to [math]\displaystyle{ i=n-r, ~ j=r-s, ~ k = s }[/math], which provides the expression given in the first paragraph.
Properties
The number of terms of an expanded trinomial is the triangular number
- [math]\displaystyle{ t_{n+1} = \frac{(n+2)(n+1)}{2}, }[/math]
where n is the exponent to which the trinomial is raised.[3]
Example
An example of a trinomial expansion with [math]\displaystyle{ n=2 }[/math] is :
[math]\displaystyle{ (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca }[/math]
See also
- Binomial expansion
- Pascal's pyramid
- Multinomial coefficient
- Trinomial triangle
References
- ↑ Koshy, Thomas (2004), Discrete Mathematics with Applications, Academic Press, p. 889, ISBN 9780080477343, https://books.google.com/books?id=90KApidK5NwC&pg=PA889.
- ↑ Harris, John; Hirst, Jeffry L.; Mossinghoff, Michael (2009), Combinatorics and Graph Theory, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 146, ISBN 9780387797113, https://books.google.com/books?id=DfcQaZKUVLwC&pg=PA146.
- ↑ Rosenthal, E. R. (1961), "A Pascal pyramid for trinomial coefficients", The Mathematics Teacher 54 (5): 336–338, doi:10.5951/MT.54.5.0336.
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