Multinomial Coefficients

Date: 02/02/2004 at 19:42:28
From: Igor
Subject: Finding coefficients of polynomial terms

What is the coefficient of the x^2*y^3*z^4 term in the (very poly!)
polynomial (x+y+z)^9 ?

I have figured out that if you keep the 3 y's and 4 z's at a certain
location and just rearrange the x's there are 9C2 = 36 combinations
possible and if you keep the x's and z's still there are 9C3 ways to
move around the y's and same thing for the z's but where do I go from
there? 

Please help: maybe there's some sort of theorem or something?

Date: 02/02/2004 at 21:31:24
From: Doctor Vogler
Subject: Re: Finding coeff. of term of (large polynomial) ^ large power

Hi Igor,

You seem to be familiar with the Binomial Theorem and binomial
coefficients (called "n-choose-r").  What you need here is called
"multinomial coefficients."   

Basically, if you have a sum of k variables (in your case, k=3), and
you raise it to a power n (in your case, n=9), then you can calculate
the coefficient on any particular term as follows:

  Notice that the sum of the exponents in any term will be n
  (as it is in your example, 2+3+4=9).  Take the factorials
  of each of the exponents, multiply them together, and
  divide n! by the product.  That is your coefficient.  (Note
  how when k=2, you get the binomial coefficient.)

For example, if you have three variables,

  (x + y + z)^n,

then the coefficient of the

  x^i * y^j * z^k

term (where i+j+k = n) is

     n!
  --------
  i! j! k!

Let me know if you'd like to talk about this some more, or if you have
any other questions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/