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# Discrete Mathematics Warmups

If there are only a handful of objects, then you can count them with a moment's thought, but the techniques of combinatorics can extend to quickly and efficiently tabulating astronomical quantities.

# Discrete Mathematics Warmups: Level 4 Challenges

Three points are chosen randomly from the circumference of a circle. What is the probability that they lie on a common semicircle?

Bonus: Generalize this to $$n$$ points.

$$20$$ men are held captive by a pirate lord, including two friends Jack and Tony.

The pirate decides to set $$10$$ of them free. The $$20$$ men are randomly divided into $$10$$ pairs. Each pair of men then flip a fair coin to decide who goes free.

The probability that both Jack and Tony are set free is $$\frac{A}{B}$$ where $$A$$ and $$B$$ are co-prime positive integers. Find the value of $$A+B.$$

Consider a $$100$$-sided polygon. If you join any $$4$$ of the $$100$$ vertices of the polygon, you get a quadrilateral.

How many quadrilaterals can be formed without including the sides of the $$100$$-sided polygon?

There is a combination safe with four switches on the front, each with three positions – low, medium, and high. There are $$3^{4} = 81$$ possible combinations.

However, this is a cheap safe and only two of the switches actually matter. If you set those two switches right, the safe will open. You do not know which are the important switches or which positions work. What is the minimum number of combinations you must try to guarantee that you will open the safe?

There are 10 stations on a circular path. A train has to stop at 3 stations such that no two stations are adjacent. How many such selections are possible?

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