Find the number of ways of arranging 10 balls (5 similar, 5 distinct) in 4 different boxes such that each box gets at least 1 similar ball and at least 1 distinct ball? Here identical ball arrangement can be done in 4 ways. whereas the distinct ball arrangement is done in 5c4 *4!*4. So total number of ways im getting is 5c4 *4!*4 *4. But there is no option supporting this where am i going wrong.

Hi,

The problem with this approach is that there will be repetitions when you do 5C4 * 4! * 4.

For example, you select A, B, C, D and put them in 1, 2, 3, 4 respectively. Say E goes into 1.
In another selection, it could be that you are selecting E, B, C, D, putting them in 1, 2, 3 and 4 and then putting A in 1. These two are the same arrangements but you are counting them twice.

The formula for onto functions from a set of m elements to a set of n elements is:

summation of $$(-1)^k ^nC_k (n-k)^m$$ where k goes from 0 to n