Determine the minimum number of people in a group to guarant

Determine the minimum number of people in a group to guarantee that at least three have the same birthday if

(a) February 29 is a valid birthday.

(b) no one was born in a leap year.

Solution

(a)

If February 29 is considered to be a valid birthday, then we have total 366 unique choices for birthdays. Suppose, we have 366 people whose birthdays are unique(i.e. no two people have the same birthday). If we consider two more people having the same birthday (means these two people were born on the same day of the year), then we can find one person among the earlier chosen 366 persons whose birthday matches with that of these two people. Hence, in order to ensure that at least 3 people have same birthday, we must choose 366+2=368 people.

(b)

If none was born in a leap year, then we have total 365 unique choices for birthdays. Suppose, we have 365 people whose birthdays are unique(i.e. no two people have the same birthday). If we consider two more people having the same birthday (means these two people were born on the same day of the year), then we can find one person among the earlier chosen 365 persons whose birthday matches with that of these two people. Hence, in order to ensure that at least 3 people have same birthday, we must choose 365+2=367 people