Question relating to probablilties

[adeel]

I have a test tomorrow and there are 2 question I am stuck on

Five patrons check their coats at a theatre. In the confusion at the end of the performance, the attendant, in despair, hands the coats back at random

What is the probability taht 3 or more of the patrons will get their coat back.

The hint given is: can exactly 4 people receive their correct coat? I am pretty sure they can't because to do that, the fifth person would get the right coat back as well.

second one:

You are doing some wordprocessing, and you accidentally type 3 letters if the alphabet without noticing which ones you typed. What is the probability that the letters will be in alphabetical order from left to right?

So can anyone help? I knoe what the asnwers are, i just dotn know how to get them (the answer is 11/120 and 1/6)

someone, please explain!

 

[metacristi]

1.

The hint given is: can exactly 4 people receive their correct coat? I am pretty sure they can't because to do that, the fifth person would get the right coat back as well.

This means that there is a single case when 4 or more of the patrons will get their coat back.

In how many ways can the attendant hand back the coats?

N=permutations of 5=5!=120

Next it must be calculated the number of possible cases when the attendant hands exactly 3 correct coats back.

N[3]=C₅³=5!/(3!)*(5-3)!=10

The probability that 3 or more of the patrons will get their coat back is:

P=(N[3]+1)/120=11/120


2.I assume that the three letters are distinct.Whatever letters are typed we can form 3!=6 distinct possibilities with them,function of their position (on the first,second or third place).

For example if the three letters were a,d,e we have the possibilities:

a,d,e
a,e,d
d,e,a
d,a,e
e,a,d
e,d,a

In how many cases are they in the alphabetical order?In exactly one case.

Hence the seeked probability is:P=1/6.

The same is valid no matter the letters written,assuming they are distinct.