Posted by Jonathan Burros on November 12, 2002 at 12:02:30:
In Reply to: expanding the binomial expansion posted by Gordon on November 08, 2002 at 22:57:08:
: Is there any type of formulas or patterns for (a+b+c)^n, (a+b+c+d)^n,...?
There is a formula called the Multinomial
Theorem
(a+b+c)^n =Sum[n!/[(n1!)(n2!)(n3!)]a^n1 b^n2 c^n2]
Where n1 + n2 + n3 = n
For (a+b+c+d)^n
Find all natural numbers n1,n2,n3,n4 such that
n1 + n2+ n3 + n4 = n
For each such 4 tuple (n1,n2,n3,n4)
form the term
n!/[n1! n2! n3! n4!] a^n1 b^n2 c^n3 d^n4
If you add up all these terms
you get the expansion of (a+b+c+d)^n