Question

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?

$2$

$3$

$4$

$5$

$12$

Answer: D

Concepts:pigeonhole principle

Difficulty rating: 1070

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 .

2011 AMC 10B Problem 11

Answer: D

Difficulty rating: 1070

Solution:

Problem 11 in Other Years