Textbook Problem

Assume that birthdays are equally likely to occur in any one of the 12 months of the year.

a. Given a group of four people, A, B, C, and D, what is the total number of ways in which birth months could be associated with A, B, C, and D? (For instance, A and B might have been born in May, C in September, and D in February. As another example, A might have been born in January, B in June, C in March, and D in October.)

b. How many ways could birth months be associated with A, B, C, and D so that no two people would share the same birth month?

c. How many ways could birth months be associated with A, B, C, and D so that at least two people would share the same birth month?

d. What is the probability that at least two people out of A, B, C, and D share the same birth month?

e. How large must n be so that in any group of n people, the probability that two or more share the same birth month is at least 50%?