Question

Suppose 7 students take a test with 4 problems, of which each student is required to submit solutions to exactly 2 problems. How can I show there are at least 2 students that solve the same 2 problems?

Answer 1

There are 4C2 ways to select 2 problems out of 4.
4C2 = 4! / (2! * 2!) = 4*3/2 = 6.

So if there are 7 students total, and there are just 6 total combinations, at least one of them has got to repeat.

Answer 2

A slightly longer method.
Let the four questions be A, B, C and D.
These are the various possibilities of choosing 2 questions out of 4.
AB
AC
AD
BC
BD
CD

Assign each possibility to just one student. 6 possibilities will be assigned to 6 students.
The seventh student has to be assigned one possibility which will be a repeat of the possibility assigned to someone else.

Answer 3

What does the C mean? @aum

Answer 4

ohh k

Answer 5

C: Combinations. P: Permutations.

Answer 6

yes i know what you mean now @aum

Answer 7

thanks!

Answer 8

you are welcome.