Question

There are $5$ people and $5$ different pairs of shoes. If each person chooses at random one right shoe and one left shoe, then the number of combinations that can be made in which no one gets a correct pair is

• A. $44$
• B. $120$
• C. $2880$
• D. $5280$

Answer 1

The correct option is D $5280$

There are $5!$ ways they can choose the $5$ right shoes (one for each) out of $5$ pairs.
For each of those $5!$ ways of choosing a right shoe, we need to make derangement of remaining $5$ left shoes so that there should not be a correct pair.
Derangement of $5$ pairs can be done in $44$ ways as $D_5=44$

Now, the required number of ways
$=5!\times D_5=44\times 120=5280$