1. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable and neither are the boxes?
2. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable but the boxes are?
1. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable and neither are the boxes?
Assuming that we can have empty boxes.....this boils down to the number of ways that we partition 6 identical balls into 3 identical boxes
These are
6, 0, 0
5, 1, 0
4, 2, 0
4, 1, 1
3, 3, 0
3, 2, 1
2, 2, 2
So......7 ways
2. How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable but the boxes are?
Again....assuming that we can have empty boxes.....the number of ways of distributing k identical balls into n distinguishable boxes is given by :
C ( k +n - 1 , n - 1) = C (6 +3 - 1, 3 - 1 ) = C (8,2) = 28 ways