Probability Homework Help!!?

[philip c]

Suppose 5 men enter a restaurant and each of them gives his hat to a receptionist. If the receptionist randomly passes the 5 hats back to the 5 men when they leave,

a) what is the probability that each man gets the correct hat?
b) Suppose now that only 3 of the 5 men have hats when they enter, and that the receptionist gives the 3 hats back to 3 randomly picked men (from the 5) when they leave. What is the probability that each of the 3 hats will be returned to the correct owner?

[holdm]

there are 5! ways to distribute the hats. only one is correct.
1/120

there are 5C3 = 10 ways to choose the three who get hats. then there are 3! = 6 ways to distribute the hats among those three. 1/(10*6) = 1/60

[rmtzlr]

a)the probability that each man gets his correct hat back is found by finding how many different ways you can distribute the hats. This turns out to be

5! = 120

this means the probability of giving each man his correct hat is:

1/120 = 0.0083

This is because it is clear that there is only one way to give back each hat to its correct owner.

b) In this case, you need to know the probability that the three men with hats receive hats, and the probability that these three men receive their correct hat. The probability that the hostess gives the 3 hats to men whom had hats is also found using combinatorics. We find the number of ways 3 men can be selected, which is 5choose3:

5!/(2!*3!) = 10

The probability that the 3 men who had hats get hats is then 1/10.
The probability that the three men get there correct hat back given that the three men receive hats is found the same way as we found part a. The number of ways that the hats can be distributed is equal to:

3! = 6

so the probability they get their correct hat given they received hats is 1/6.

This means that the probability that each of the three men get their correct hat back is:

(1/10)*(1/6) = 1/60 = 0.0167