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

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 number of terms of an expanded trinomial is the triangular number

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

See also[edit]


  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 .