Posted by Joe on Monday, May 23, 2011 at 9:27pm.
The number of terms in a multinomial expansion of m terms raised to the nth power is equal to C(n+m-1,n).
In the present case,
n=5, m=3, so
C(n+m-1,n)=C(5+3-1,5)=C(7,5)=7*6/(2*1)=21
In the same vein, you may be interested to know that the individual coefficients for the terms
a^p*b^q*c^r (where p+q+r=n)
is
C(n,p,q,r)
=n!/(p!q!r!)
For example, the term a²b²c
has p=2,q=2,r=1
so the coefficient is
C(5,2,2,1)
=5!/(2!2!1!)
For more interesting facts, see:
http://en.wikipedia.org/wiki/Multinomial_theorem