Question

Put 5 distinguishable balls in 4 boxes. List all the possibilities of different arrangements of balls in 3 boxes and find the most probable distribution.

Answer 1

According to the question, we have 5 balls to be placed in 3 boxes where no box remains empty.

Hence, we can have the following kinds of distributions:

1. Firstly, where the distribution will be (3,1,1) that is, one box gets three balls and the remaining two boxes get one ball each.
   • No. of ways to choose the box which gets 3 balls = 3 (since, 3 boxes are there).
   • No. of ways to select 3 balls out of 5 is =
   • No. of ways to distribute rest of the balls =2.
   • Total no. of ways =60

2. Secondly, where the distribution will be (1,2,2) that is, one box gets one ball and the remaining two boxes get two balls each.

• No. of ways to select 1 ball out of 5 is =
• No. of ways to choose the box which gets 1 balls = 3 (since 3 boxes are there).
• No. of ways to select 2 balls out of remaining 4 to go in the second box is = .
• Total no. of ways = 90

Total:60+90=150.