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

${\displaystyle (a+b+c)^{n}=\sum _{\stackrel {i,j,k}{i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},}$

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

${\displaystyle {n \choose i,j,k}={\frac {n!}{i!\,j!\,k!}}\,.}$

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]

Properties

The number of terms of an expanded trinomial is the triangular number

${\displaystyle t_{n+1}={\frac {(n+2)(n+1)}{2}},}$

where n is the exponent to which the trinomial is raised.[3]

Example

An example of a trinomial expansion with ${\displaystyle n=2}$ is :

${\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca}$

See also

• Binomial expansion
• Pascal's pyramid
• Multinomial coefficient
• Trinomial triangle

References

1. Koshy, Thomas (2004), Discrete Mathematics with Applications, Academic Press, p. 889, ISBN 9780080477343.
2. Harris, John; Hirst, Jeffry L.; Mossinghoff, Michael (2009), Combinatorics and Graph Theory, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 146, ISBN 9780387797113.
3. Rosenthal, E. R. (1961), "A Pascal pyramid for trinomial coefficients", The Mathematics Teacher, 54 (5): 336–338.