Letters in box?

Question

Let's say you have two boxes, each with the letters A and B in them. You pick one letter from each box - have a total of four combinations (AA, AB, BA, BB). But, let's say you only want unique combinations (order does not matter), so there are really only 3 combinations (AB and BA are the same).

You can do the same with two boxes with the letters A, B and C in them. There are 9 total combinations, but only 6 unique combinations.

Is there a general way to figure this out regardless of the number of boxes or letters in the boxes?

Swamy - Neither the combinations nor permutations equations work for the above examples

I think this is the answer I'm looking for:

b= number of boxes
k = number of letters

(b+k-2)C(k-2) + (b+k-2)C(k-1)

where C is the choose/combination function.

Is that correct?

Answer 1

Let's try and think about this a little.

If you had 1 box with k letters, then the answer is 1, 2, 3, 4, 5...
If you had 2 boxes with k letters, then the answer is 1, 3, 6, 10, 15...
If you had 3 boxes with k letters, then the answer is 1, 4, 10, 20...

Hmm... these numbers remind me of Pascal's triangle and binomial distribution if you look along the diagonals.

............ 1
......... 1 .. 1
...... 1... 2 .. 1
... 1... 3 .. 3 .. 1
. 1 . 4 .. 6 .. 4 .. 1
1. 5 .10 .10 . 5 .. 1

................... n!
C(n,k) = -----------
.............. k! (n-k)!

Answer 2

n^2-n for n boxes.
2^2-2 =2
3^2-3 =9-3=6

Answer 3

Yes, what you are asking is the theory of permutations and combinations. Just type permutations and combinations in yahoo or google search and you will get several websites for help.