Discrete Mathematics

Discrete Mathematics Warmups

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 nn points.

2020 men are held captive by a pirate lord, including two friends Jack and Tony.

The pirate decides to set 1010 of them free. The 2020 men are randomly divided into 1010 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 AB\frac{A}{B} where AA and BB are co-prime positive integers. Find the value of A+B.A+B.

Consider a 100100-sided polygon. If you join any 44 of the 100100 vertices of the polygon, you get a quadrilateral.

How many quadrilaterals can be formed without including the sides of the 100100-sided polygon?

There is a combination safe with four switches on the front, each with three positions – low, medium, and high. There are 34=813^{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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