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 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 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 trinomial expansion can be calculated by applying the binomial expansion twice, setting

in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index

The product of the two binomial coefficients is simplified by shortening

, and comparing the index combinations here with the ones in the exponents, they can be relabelled to

The number of terms of an expanded trinomial is the triangular number where n is the exponent to which the trinomial is raised.