Probability Question for NY State Teaching Exam

Question

Question: What is the probability that in a group of five friends, at least two share the same astrological sign? Note that there are 12 astrological signs of the Zodiac.

 

My Work: I have no idea how to get this answer. My first instinct is to do 5 x 12= 60 to get the set of all possibilities. But clueless as the where I go from there. I don't understand how the book arrived at this answer and why it's correct. Please explain.

 

Book Answer: 

 

1- (11 x 10 x 9 x 8)/12⁴

 

1- (7,920/20,736) = 0.38194444 = .6180556

Answer 1

P(at least 2) = 1 - P(less than 2)

 

The first friend has a 1/12 chance of having one of the signs, leaving the second friend 11/12 chance of NOT sharing. The 3rd friend has a 10/12 chance of not sharing with the previous two, etc.

 

1 - (11/12)(10/12)(9/12)(8/12) = 1 - (11 x 10 x 9 x 8)/12⁴