Lets say I have 4 balls (numbered 1, 2, 3, 4) and 15 slots.
Each of the 15 slots can hold minimum 0 balls and maximum 4 balls.
It is not necessary to place all balls in the slots. i.e. We can put only 1 ball, or only 2 balls, or even 0 balls in the 15 slots.
The order in which the balls are placed in the slots does not matter.
For example –
Slot Combin 1 Combin 2 Combin 3 Combin 4 Combin 5 Combin 6
1 1,2,3,4 1,2,3 0 1,4 1 0
2 0 0 3,1,2 0 2 0
3 0 0 0 0 3 0
4 0 0 0 0 4 0
5 0 0 0 0 0 0
6 0 0 0 2,3 0 0
7 0 0 0 0 0 0
8 0 0 0 0 0 0
9 0 0 0 0 0 0
10 0 0 0 0 0 0
11 0 0 0 0 0 0
12 0 0 0 0 0 0
13 0 0 0 0 0 0
14 0 0 0 0 0 0
15 0 4 0 0 0 0
How many unique combinations are possible? Which formula would be applicable for this scenario?
Thanks!
Best Answer
There are $15$ choices of slot for each ball, and these choices are made independently for each ball, so they can be made in $15^4=50,625$ ways. Equivalently, you’re just counting the possible $4$-tuples of slot numbers, where $\langle 1,1,1,1\rangle$ represents your first combination, $\langle 1,1,1,15\rangle$ represents your second combination, $\langle 1,6,6,1\rangle$ represents your fourth combination, and so on.