Question

Below is a list of properties that a group of people might possess. For each property, either give the minimum number of people that must be in a group to ensure that the property holds, or else indicate that the property need not hold even for arbitrarily large groups of people. (Assume that every year has exactly 365 days; ignore leap years.)

(a) At least 2 people were born on the same day of the year (ignore year of birth).

(b) At least 2 people were born on January 1.

(c) At least 3 people were born on the same day of the week.

(d) At least 4 people were born in the same month.

(e) At least 2 people were born exactly one week apart.

Answer 1

a) 366: If there are 366 students or more, since total no of days are 365, there must be atleast 2
students who is having the same day of the year.

b) Not possible: It is not necessary that atleast 2 born in January 1. Because it is possible for any

large no of students that all them born in December only, or in June only. It is possible that

none of even very large no of students have their birth on 1st Jan

c) 15: Since their are 7 days in a week, if there are 15 or more students there must be atleast three

students who is having the same day of birth.

d) 37: If there are 37 or more students, since we have only 12 months, so there must atleast 4 students

who is having the same month of birth.

e) Not possible: Whatever be the no of students, it’s not sure. For an example, each students have

birthday on 1st of every month.