# 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]

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]