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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

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

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. The trinomial coefficients are given by

${\displaystyle {n \choose i,j,k}={\frac {n!}{i!\,j!\,k!}}\,.}$

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.

Derivation

The trinomial expansion can be calculated by applying the binomial expansion twice, setting ${\displaystyle d=b+c}$, which leads to

${\displaystyle {\begin{aligned}(a+b+c)^{n}&=(a+d)^{n}=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,d^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,(b+c)^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,\sum _{s=0}^{r}{r \choose s}\,b^{r-s}\,c^{s}.\end{aligned}}}$

Above, the resulting ${\displaystyle (b+c)^{r}}$ in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index ${\displaystyle s}$.

The product of the two binomial coefficients is simplified by shortening ${\displaystyle r!}$,

${\displaystyle {n \choose r}\,{r \choose s}={\frac {n!}{r!(n-r)!}}{\frac {r!}{s!(r-s)!}}={\frac {n!}{(n-r)!(r-s)!s!}},}$

and comparing the index combinations here with the ones in the exponents, they can be relabelled to ${\displaystyle i=n-r,~j=r-s,~k=s}$, which provides the expression given in the first paragraph.

Properties

The number of terms of an expanded trinomial is the triangular number

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

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

Example

An example of a trinomial expansion with ${\displaystyle n=2}$ is :

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

See also

• Binomial expansion
• Pascal's pyramid
• Multinomial coefficient
• Trinomial triangle

References

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, doi:10.5951/MT.54.5.0336.

• Factorial and binomial topics