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_{\stackrel{i,j,k}{i+j+k=n}} {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 $i + j + k = n$ .^[1] The trinomial 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 $m = 3$ . 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

t_{n+1} = \frac{(n+2)(n+1)}{2},

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

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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[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 .

Trinomial expansion

See also

References