# Brainteaser #30- 67 Students -

In a class of 67 students, the probability that at least two of them have the same number of friends in
the class is very high. But is it a certainty (i.e. is the probability 100%)? Note that if one student
is friends with another student then that other student is also friends with him/her (i.e. friendships
are mutual). And of course no one is friends with himself/herself.

**HINT**: If one student has no friends, then no student can have 66 friends.

**ANSWER**:

*Yes, it is a certainty.*
**EXPLANATION**: The number of students in the class that any given student is friends with
will be between 0 (not friends with anyone in the class) and 66 (friends with everyone in the class).
For no two students to have the same number of friends, then each of the 67 students would have to be
friends with a unique number of students in the class (from 0 to 66). But if one of the students has
66 friends, then no student can have 0 friends so only one of those two numbers is possible. So that
means there are only 66 unique numbers for the number of friends and there are 67 students so it can be
said with 100% certainty that at least two students have the same number of friends in the class.

Do you have a

suggestion for this puzzle (e.g. something that should
be mentioned/clarified in the question or solution, bug, typo, etc.)?