vempatichaitu wrote:

in how many ways can we distribute

(1)5 different balls to 3 different baskets

(2)5 identical balls to 3 different baskets

(3)5 different balls to 3 identical baskets

(4)5 identical balls to 3 identical baskets

Lets tackle the easier question first

2) 5 identical balls to 3 different baskets

This falls into the category called Similar -> Different

As it is similar to different, we don't have to do selection as all of them are same. Only arrangement is required.

A + B + C = 5 => 7C2 = 21

(4)5 identical balls to 3 identical baskets

As again 5 identical balls, there is no selection of balls.

S -> S

Lets say 5 balls goes into the 5 1st basket but as baskets are also identical, hence it doesn't matter. - So we can differentiate based on numbers.

5, 0, 0 - 1 way

4, 1, 0 - 1 way

3, 2, 0 - 1 way

3,1,1 - 1 way

2, 2, 1 - 1 way

Hope I have counted all the possibilities - 5 ways.

(3)5 different balls to 3 identical baskets

D -> S

As the balls are different, the selection matters.

5, 0, 0 - can be only 1 way as all the balls are in a basket and baskets are all identical.

4, 1, 0 - 5C4 - 5 ways.

3, 2, 0 - 5C3 - 15 ways

3, 1, 1 - 5C3 * 2C1 = 15 * 2 = 30 ways.

2,2,1 - 5C2 * 3C2 = 15 * 3 = 45 ways.

Again I hope I have covered all the possibilities.

So total number of ways 96 ways.

(1)5 different balls to 3 different baskets

Now this is a biggie. Selection and arrangement both are required.

D-> D

5, 0, 0 -> 5C5 (Selection) * 3C1 (Arrangement) = 3 ways

4, 1, 0 -> 5C4 * 3C1 * 1C1 * 2C1 = 5 * 3 * 2 = 30 ways

3, 2, 0 -> 5C3 * 3C1 * 3C2 * 2C1 = 15 * 3 * 2 = 90 ways

3, 1, 1 -> 5C3 * 3C1 * 2C1 * 2C1 * 1C1 * 1C1 = 15 * 3 * 2 * 2 = 180 ways

2, 2, 1 -> 5C2 * 3C1 * 3C2 * 2C1 * 1C1 * 1C1 = 15 * 3 * 3 * 2 = 270 ways

Hope I have covered all the possibilities again.

573 ways.

Not sure If I have covered all the possibilities. I can bet on you that apart from 1 st one, other cases may not come in GMAT in the near future. This I learnt from CAT prep days.

the answer is going to be 3^5 ways as each ball has an option to go into any of the 3 baskets

and your answer for the case of- 5 different balls to 3 identical baskets needs to be corrected also

5, 0, 0 - can be only 1 way as all the balls are in a basket and baskets are all identical.

4, 1, 0 - 5C4 - 5 ways.

2,2,1 - 5C2 * 3C2 = 15 * 3 = 45 ways /(2!)(as these cases have baskets taking the same no.of balls)