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]

Properties

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]

Example

An example of a trinomial expansion with $n=2$ is :

$(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$

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.

[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.

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[2]

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Properties

Example

See also

References