Multinomial theorem

In mathematics, the multinomial formula is an expression of a power of a sum in terms of powers of the addends. For any positive integer m and any nonnegative integer n, the multinomial formula is

$(x_{1}+x_{2}+x_{3}+\cdots +x_{m})^{n}=\sum _{k_{1},k_{2},k_{3},\ldots ,k_{m}}{n \choose k_{1},k_{2},k_{3},\ldots ,k_{m}}x_{1}^{k_{1}}x_{2}^{k_{2}}x_{3}^{k_{3}}\cdots x_{m}^{k_{m}}$

The summation is taken over all combinations of the indices k₁ through k_m such that k₁ + k₂ + k₃ + ... + k_m = n; some or all of the indices may be zero. The numbers

${n \choose k_{1},k_{2},k_{3},\ldots ,k_{m}}={\frac {n!}{k_{1}!k_{2}!k_{3}!\cdots k_{m}!}}$

are the multinomial coefficients.

The binomial theorem and binomial coefficient are special cases, for m = 2, of the multinomial formula and multinomial coefficient, respectively.