Multinomial Theorem Proof. As a proof by induction use something like x 1 x 2 dots x n x n 1 r y x n 1 r say with y x 1 x 2 dots x n. First for k 1 k 1 both sides equal xn 1 x 1 n.
In addition we shall use multi index notation. 3 3 multinomial theorem theorem 3 3 0 for real numbers x1 x2 xm and non negative integers n r1 r2 rm the followings hold. Look it up.
The below proof of the multinomial theorem uses the binomial theorem and induction on k k.
First for m 1 both sides equal x 1 n since there is only one term k 1 n in the sum. For the induction step suppose the multinomial theorem holds for k k. The algebraic proof is presented first. By the binomial theorem this becomes.
