Multinomial Theorem Expansion. Especially when dealing with multinomials it is expedient to check whether we have forgotten any terms by adding up the coefficients and also checking the expected sum of the coefficients in each group. It expresses a power.
We will show how it works for a trinomial. For any positive integer m and any nonnegative integer n the multinomial formula tells us how a sum with m terms expands when raised to an arbitrary power n. The binomial theorem gives us as an expansion of x y n.
It describes the result of expanding a power of a multinomial.
X 1 x 2 x m n k 1 k 2 k m n n k 1 k 2 k m t 1 m x t k t displaystyle x 1 x 2 cdots x m n sum k 1 k 2 cdots k m n n choose k 1 k 2 ldots k m prod t 1 m x t k t. Consider a b c 4. The multinomial theorem provides a formula for expanding an expression such as x1 x2 xk n for integer values of n. The multinomial theorem below provides this formula as an extension to the previous two theorems.
