Question

A group of 5 men are meeting for lunch.

(a) Each of the 5 men is wearing a coat when they enter the restaurant. Upon arrival, they each give their coat to the hostess. However, the hostess forgets to take note of which coat belongs to which man so she decides to guess. If the hostess randomly passes the 5 coats back to the 5 men when they leave, what is the probability that each man gets the correct coat?

(b) Suppose now that only 2 of the 5 men is wearing a coat when they enter. Again, the hostess is forgetful and cannot remember which 2 men were wearing coats, much less which coat belongs to each of these 2 men. So, once again, she must guess. When they are ready to leave, the hostess randomly picks 2 men who she believes were wearing coats and then gives the 2 coats back to these 2 randomly picked men. What is the probability that each of the 2 coats will be returned to the correct owner? (In other words, the hostess has to first pick the correct 2 men from the group of 5 men, and then she has to give the correct coats back to each of these 2 men.)

Answer 1

Tile:

See the explanation.

Step-by-step explanation:

(a)

There were total 5 men wearing coats.

5 coats can be returned to 5 men in 5! = 120 ways.

The coats can be returned to the accurate persons only in 1 way.

Hence, the probability that each man gets the correct coat is $\frac{1}{120}$.

(b)

At the time of returning the first coat, the hostess will have 5 choices and for the second she will have 4 choices.

Hence, in $5\times4 = 20$ ways the hostess can return the 2 coats.

There is only 1 possible case that each of the coats will return to the correct owner.

Hence, the required probability is $\frac{1}{20}$.