Binomial Coefficients and Generating Functions - Cs.ioc.ee

Number of solutions of the equation x1 + x2 + ··· + xk = n
Theorem
The number of ways to distribute n identical objects
into k bins is n+k−1
n .
Proof.
The number of ways to distribute n objects equals to the number of solutions of
x1 + x2 + ··· + xk = n that is coefficient of x n of the generating function
G(x) = 1/(1 − x) k = (1 − x) −k .
For recollection: Maclaurin series (a Taylor series
expansion of a function about 0):
.
f (x) = f (0) + f ′ (0)x + f ′′ (0)
2! x2 + f ′′′ (0)
x
3!
3 + ··· + f (n) (0)
x
n!
n + ···