Putting balls in boxes

Probability Level 3

How many ways are there to place 4 4 distinct balls into 5 5 distinct boxes such that exactly 3 3 of the 5 5 boxes do not have any balls?


The answer is 140.

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Lawahar Qasid by Lawahar Qasid

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3 solutions

Note that exactly two boxes are going to be filled.

Call them A and B.

Case I - There are 3 balls in A and 1 in B

This can be thought of as choosing which ball that would be there in B. This can thus be done in 4 ways

Case II - There are 1 balls in A and 3 in B

This can also be done on 4 ways.

Case III - There are two balls in A and two in B.

Think of this as choosing two balls which shall go to B. There are 6 ways.

Thus there are a total of 14 ways to distribute the balls in the two boxes.

Also, there are (5 choose 2) or 10 ways to choose the two boxes out of all five.

Thus, you can do it in 140 ways

6 Helpful 0 Interesting 0 Brilliant 0 Confused

First let's number the boxes: 1, 2, 3, 4 & 5 and the balls: 1, 2, 3 & 4. There are only two cases: * Case I: there are 2 balls in a box and the other 2 in another box: - Balls 1 & 2 together and balls 3 & 4 together: there are 5 chances for balls 1 & 2: boxes 1, 2, 3, 4 & 5. with each of them, there are 4 chances for balls 3 & 4 to fill the left 4 boxes. So, the total probabilities in this way= 5 * 4 = 20. - Balls 1 & 3 together and balls 2 & 4 together: 5 * 4 = 20. - Balls 1 & 4 together and balls 2 & 3 together: 5 * 4 = 20. So, the total ways for this case= 20 * 3 = 60. * Case I: there are 3 balls in a box and 1 ball in another box: - Balls 1, 2 & 3 together and balls 4 alone: There are 5 chances for balls 1, 2 & 3: boxes 1, 2, 3, 4 & 5. with each of them, there are 4 chances for ball 4 to fill the left 4 boxes. So, the total probabilities in this way= 5 * 4 = 20. - Balls 1, 2 & 4 together and balls 3 alone: 5 * 4 = 20. - Balls 1,3 & 4 together and balls 2 alone: 5 * 4 = 20. - Balls 2, 3 & 4 together and balls 1 alone: 5 * 4 = 20. So, the total ways for this case= 20 * 4 = 80. So, the total ways for both cases= 60 + 80 = 140.

د. خالد لاشين - 6 years, 12 months ago
Avraam Aneleitos
Jan 3, 2015

There are 5!/((5-2)!2!)=10 ways to choose the two boxes that are going to be filled.Then the number of possible ways to place the four balls in the two boxes is 2^4 reduced by two since two of those ways have all the balls in one box leaving four boxes empty while we want them to be three.Thus the number of ways this can be done is 10*14=140.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Anna Anant
Jan 24, 2015

5C3 (2^4-2)... 10 14=140....(Ans.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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