pigeonhole principle - 15 subjects, 18 students, each student takes 4 subjects. Prove there are 2 students with same 2 subjects.

Question

There are 15 subjects in a school. Every student takes 4 subjects out of them. Given that the school has 18 students, prove that 2 of the students have 2 common subjects.

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I got the number of subject combinations being 15C4 = 1365. But I don't see how this is helpful as we are only interested in two students having two common subjects. I know the 'pigeons' in this case would be the number of students, but I don't know what the 'pigeonholes' would be (which I supposed would be less than 18 then we can prove it is true using the principle).

Answer 1

First of all, you shouldn't start writing combinations without knowing what are the elements you're combining. A better way to start the proving is assuming the initial statement is false.
Hint: Use the pigeonhole principle and you will get a contradiction assigning each student different subjects avoiding the case 2 of them have the same 2 subjects.
For example:
Student A have subjects (1,2,3,4)
Student B have subjects (5,6,7,8)
SC have (9,10,11,12)
SD have (13,14,15,1)
SE have (2,6,9,13)
SF have (3,7,10,14)
...
And eventually you'll run out of subjects without more than 2 students.

Answer 2

There are 105 pairs of subjects. Each of the 18 students chooses 6 pairs, which makes 108 pairs.