2011 AMC 10B Problem 11

Problem 11 of 25IntermediateCombinatorics

There are

52

people in a room. What is the largest value of

n

such that the statement “At least

n

people in this room have birthdays falling in the same month” is always true?

Show solution

Solution

It isn’t necessarily true for

n\geq 6

as we could have

5

people born in the first

4

months and

4

people born in the subsequent months. However, one month must be greater than or equal to

5

as the average of the number of people born in each month is

\frac{52}{12}

which is greater than

4,

and some month must be above average. Thus, the correct answer is D .

Tagged: pigeonhole principle