Have another question for you today:
A course has seven elective topics, and students must complete exactly three of them in order to pass the course. If 200 students passed the course, show that at least 6 of them must have completed the same electives as each other.
Now I know this is related to counting and the pigeonhole principle, and there are a couple of other related questions already asked but I couldn't apply them to my quuestion.
I know that the (informal) pigeonhole principle states that if you have $n$ boxes, and you have more than $n$ pigeons to distribute between those boxes, then at least one of the boxes will contain more than one pigeons, but I'm not sure what my boxes and what my pigeons are in this problem
Best Answer
Let us first count the number of topic choices. There are 7 course and a student must choose 3 among them. Therefore there are $$ {7 \choose 3} = 35$$ such choices.
Now there are 200 students. Suppose that there are at most 5 students that have completed the same electives as each other. We have $$5\times 35 =175.$$ But $175<200$ and therefore at least 6 students have completed the same electives.