# 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 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

## References

- ↑ Koshy, Thomas (2004),
*Discrete Mathematics with Applications*, Academic Press, p. 889, ISBN 9780080477343. - ↑ Harris, John; Hirst, Jeffry L.; Mossinghoff, Michael (2009),
*Combinatorics and Graph Theory*, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 146, ISBN 9780387797113. - ↑ Rosenthal, E. R. (1961), "A Pascal pyramid for trinomial coefficients",
*The Mathematics Teacher*,**54**(5): 336–338.

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