avaneeshvyas wrote:

Five balls of different colors are to be placed in three different boxes such that any box contains at least 1 ball . What is the maximum number of different ways in which this can be done?

A. 60

B. 90

C. 120

D. 150

E. 180

Please provide a small note of explanation for all the combinations used in the solution.

Since 5 balls are to be arranged in 3 different boxes and each box should contain at least 1 ball.

This can be done as 2 cases

Case 1 : 3-1-1 i.e. 3 balls in 1 box, 1 ball in 2nd box & 1 ball in 3rd box.

To count the number of different ways for this case, we will go step by step.

1. Select 3 balls from 5 balls which will be in 1 box. = 5C3

2. Naturally the other 2 balls will go in the left 2 boxes.

3. Now this ball arrangement can itself be arranged in 3! ways.

4. Total ways = 5C3 * 3! = 5*4/2/1 * 3*2*1 = 60 ways.

Case 2 : 2-2-1 i.e. 2 balls in 1 box, 2 balls in 2nd box & 1 ball in 3rd box.

Again we will go step by step for this case.

1. Select 2 balls from 5 balls which will be in 1 box. = 5C2

2. Select 2 balls from 3 left balls which will be in 2 box. = 3C2

3. The left 1 ball will be placed in 3rd box.

Total such selection = 5C2*3C2*1 /2 (since 2 ball selected in 1st and 2 ball selected in 2nd will be same in different conditions.)

4. Now this ball arrangement can itself be arranged in 3! ways.

5. total ways = (5C2 * 3C2 * 1/2) * 3!

= 90

Total ways = 60 +90 = 150 ways,

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