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Question 8 : Five racquets needs to be placed in three boxes. Each box can hold all the five racquets. In how many ways can the racquets be placed in the boxes so that no box can be empty if all racquets are different but all boxes are identical?

  1. 24
  2. 25
  3. 27
  4. 26

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Explanatory Answer

Let the 5 racquets be a, b, c, d, e
3 Boxes are Identical
Since the boxes are identical 3 in First box, 1 in Second and 1 in third is not different from 1 in first, 3 in second and 1 in third.

Box 1 Box 2 Box 3
3 racquets
5C3 ways
1 racquets
2C1 ways
1 racquets
1C1 way
2 racquets
5C2 ways
2 racquets
3C2 ways
1 racquets
1C1 way

But there is a catch in the above table.
Imagine this process, after selecting 3 out of a, b, c, d, e
Let's say we select d, e, c. a and b have to be put in two identical boxes.
This can be done only in one way.

In second scenario let's assume a and b in First box and c and d in second box and e in third box. But the possibility will be same when c and d in First box and a and b in second box.
So, it is \\frac{^{5}C_{2} × ^{3}C_{2}}{2}) = 15 ways, In first scenario 5C3 = 10 ways
Totally 25 ways.

The answer is 25

Choice B is the correct answer.


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