Trinomial expansion

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In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by

$(a+b+c)^n = \sum_{i,j,k} {n \choose i,j,k}\, a^i \, b^j \, c^k$

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. The coefficients are given by

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

This formula is a special case of the multinomial formula for m = 3. It is of some interest that the coefficients are given by 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

$\frac{(n+2)(n+1)}{2}$