Question

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

• 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, required number of ways $=5!\times D_5$
$\Rightarrow 44\times 120$
$\Rightarrow 5280$