Disclaimer: Here is an amateur attempt to share my understanding regarding various Distributions in Permutations and Combinations. In case you find any errors/loopholes, please let me know.

With respect to Distribution questions in Permutation and Combinations, most of the question types can be grouped under the following categories,

1. Distribution of Similar Objects to Similar Groups

2. Distribution of Similar Objects to Different Groups

3. Distribution of Different Objects to Different Groups

4. Distribution of Different Objects to Similar Groups

Let me take an example to explain all these four types.

How do you distribute five apples to three similar baskets?

In this case, since the baskets are similar there is no need for any arrangement. We just need to frame the various cases. Arrangement within the groups is not required since the groups are all similar. So the answer would be,

(5,0,0),(4,1,0),(3,2,0),(3,1,1),(2,2,1) => 5 cases

How do you distribute five similar apples to three different baskets?

In this case, the baskets are different. Hence Arrangement of fruits within the baskets is equally important.

Consider you are distributing the apples like this,

0 0 | 0 0 | 0

By moving the sticks to various places, you can get various possibilities.

So Number of cases = 7! / 5!*2! = 7C2 = 21 ways

How do you distribute five different apples to three different baskets?

The first apple can be placed in 3 ways.

The second apple can be placed in 3 ways.

Hence All 5 apples can be placed in 3 different baskets in 3^5 ways.

How do you distribute five different apples to three similar baskets?

This is the scenario wherein many of us make errors. D-> S Scenario is ideally the application of “Combinations”. This question can be solved as follows. The various cases are,

(5,0,0)=> 5C5 => 1 way

(4,1,0) => 5C4=> 5 ways

(3,2,0) => 5C3 =>10 ways

(3,1,1) => (5C3*2C1)/2! =>10 ways

(2,2,1) => (5C2*3C2)/2! => 15 ways

Total => 41 ways

In the last two cases, we divide by 2! since we have identical groups. In case there are three identical groups we need to divide by 3!