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Problem_Set_05 - ArsDigita University Month 2 Discrete...

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ArsDigita University Month 2: Discrete Mathematics - Professor Shai Simonson Problem Set 5 – Combinatorics and Counting 1. Given ten points in the plane with no three collinear, a. How many different segments joining two points are there? b. How many ways are there to choose a directed path of length two through three distinct points? c. How many different triangles are there? d. How many ways are there to choose 4 segments? e. If you choose 4 segments at random, what is the chance that some three form a triangle? 2. Forty equally skilled teams play a tournament in which every team plays every other team exactly once, and there are no ties. a. How many different games were played? b. How many different possible outcomes for these games are there? c. How many different ways are there for each team to win a different number of games? 3. Let C(n,k) be the number of ways to choose k objects from a set of n. Prove by a combinatorial argument: a. C(n,0) + C(n, 1) + … + C(n, n) = 2 n .
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