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

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

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

Where n is the exponent to which the trinomial is raised.