multinomial theorem (proof)
Proof. The below proof of the multinomial theorem uses
the binomial theorem and induction
on .
In addition
, we shall use multi-index notation.
First,
for , both sides equal . For the induction step,
suppose the multinomial theorem holds for .
Then the binomial theorem and the induction assumption yield
where and is a multi-index in . To complete the proof, we need to show that
the sets
are equal. The inclusion is clear since
For , suppose , and . Let . Then , so for some . It follows that that .
Let us define and let be a multi-index in . Then
This completes the proof.
Title | multinomial theorem (proof) |
---|---|
Canonical name | MultinomialTheoremproof |
Date of creation | 2013-03-22 13:41:55 |
Last modified on | 2013-03-22 13:41:55 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 4 |
Author | Koro (127) |
Entry type | Proof |
Classification | msc 05A10 |