Probability

# Combinations: Level 3 Challenges

How many triangles are in a fully connected pentagon with all its diagonals?

Suppose of all possible ways to choose two representatives among 10 students in a class, there are exactly $30$ different ways to choose one or more female as a representative. How many males are there in the class?

Abe, Raith and Ghost are to commit 7 sins on earth altogether. But they can't remember all of the types, so they agree that one sin is as good as any other. If they each want to commit at least one sin on earth, how many different ways can they split up the work?

I was on vacation in England, and wanted to visit the Tower of London. The roads were laid out on a grid map, and and the castle was 4 blocks north and 5 blocks east. I also want to get at least one ice cream cone along the way. I knew that there were 2 ice cream shops that I could use, and they were conveniently located at an intersection.

If I were to travel only north and east, how many routes did I have to get to the castle?

How many ways are there to split 10 people into 2 teams so that they can play 5 on 5 basketball?

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