Multinomial Theorem

The multinomial theorem describes how to expand a power of a sum of several terms, generalizing the binomial theorem from two terms to any number of terms. For a sum of m terms raised to the nth power, it expresses the expansion as a sum of terms with multinomial coefficients of the form n factorial divided by the product of the factorials of the individual exponents, which must themselves sum to n, and the theorem reduces to the ordinary binomial theorem when m equals 2; it has applications in combinatorics, probability and polynomial algebra, including counting arrangements of objects into bins and permutations of words with repeated letters.