IPMAT Rohtak 2020 (QA) - 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? | PYQs + Solutions

IPMAT Rohtak 2020

Modern Math

Permutation & Combination

Medium

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?

Correct Option: 2

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.

\begin{array}{|c|c|c|} \hline \text{Box 1} & \text{Box 2} & \text{Box 3} \\ \hline 3 \text{ racquets} & 1 \text{ racquet} & 1 \text{ racquet} \\ 5C_3 \text{ ways} & 2C_1 \text{ ways} & 1C_1 \text{ way} \\ \hline 2 \text{ racquets} & 2 \text{ racquets} & 1 \text{ racquet} \\ 5C_2 \text{ ways} & 3C_2 \text{ ways} & 1C_1 \text{ way} \\ \hline \end{array}

But there is a catch in the above table.

\newline

Imagine this process, after selecting 3 out of a, b, c, d, e

\newline

Let's say we select d, e, c. Then a and b have to be put in two identical boxes.

\newline

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{^5C_2 \times ^3C_2}{2} = 15

ways, In first scenario

^5C_3 = 10

ways

Total

=25

ways.

IPMAT Rohtak 2020 (QA) PYQs

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?

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