Trinomial expansion

Layers of Pascal's pyramid derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial – the number of terms is clearly a triangular number

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_{\binom{i, j, k}{i + j + k = n}} (\binom{n}{i, j, k}) a^{i} b^{j} c^{k},

where $n$ Script error: No such module "Check for unknown parameters". is a nonnegative integer and the sum is taken over all combinations of nonnegative indices $i, j,$ Script error: No such module "Check for unknown parameters". and $k$ Script error: No such module "Check for unknown parameters". such that $i + j + k = n$ Script error: No such module "Check for unknown parameters"..^[1] The trinomial coefficients are given by

(\binom{n}{i, j, k}) = \frac{n!}{i! j! k!} .

This formula is a special case of the multinomial formula for $m = 3$ Script error: No such module "Check for unknown parameters".. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.^[2]

Derivation

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

\begin{aligned} (a + b + c)^{n} & = (a + d)^{n} = \sum_{r = 0}^{n} (\binom{n}{r}) a^{n - r} d^{r} \\ = \sum_{r = 0}^{n} (\binom{n}{r}) a^{n - r} (b + c)^{r} \\ = \sum_{r = 0}^{n} (\binom{n}{r}) a^{n - r} \sum_{s = 0}^{r} (\binom{r}{s}) b^{r - s} c^{s} . \end{aligned}

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

The product of the two binomial coefficients is simplified by shortening $r!$ ,

(\binom{n}{r}) (\binom{r}{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 $i = n - r, j = r - s, k = s$ , which provides the expression given in the first paragraph.

Properties

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

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

where $n$ Script error: No such module "Check for unknown parameters". is the exponent to which the trinomial is raised.^[3]

Example

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

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

References

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Trinomial expansion

Contents

Derivation

Properties

Example

See also

References

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Trinomial expansion

Derivation

Properties

Example

See also

References

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