Question

The no.of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty?

Answer 1

Represent the question as an equation. Let there are x balls in box 1, y in box 2 and z in box 3. We have:
x+y+z=8
and x>0,y>0,z>0

We need to find the number of solutions to this equation. Such equation is called the number of compositions of N into K parts. The solution to this is: CN−1k−1
For our case, N = 8 and k = 3
Thus the solution is C[7,2]=42/2=21
(8−1)C(3−1)=7C2

⇒7!/2!5!

⇒7×6×5!/2×5!

⇒21