Solution: 2013 Fall Midterm - 6

Author: Pat Morin

Question

Each person in a group of $ n $ people has a last name consisting of two uppercase letters. For what values of $ n $ can we guarantee that there are at least two people with the same last name?

(a)

$n \geq 52$

(b)

$n \geq 676$

(c)

$n \geq 677$

(d)

$n \geq 26$

Solution

There are $ 26^2 = 676 $ possible unique last names.

If $ n = 676+1 $, then we can guarantee that in the worst case scenario where the first 676 people all have different last names, the 677th person needs to pick a last name that’s already been taken