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Problem_Set_06 - ArsDigita University Month 2 Discrete...

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ArsDigita University Month 2: Discrete Mathematics - Professor Shai Simonson Problem Set 6 – Combinatorics and Discrete Probability 1. Assume someone is throwing three dice of different colors. a. How many ways are there to roll the dice? b. Make a chart showing how many of these rolls, have i as the second highest and j as the highest value, for 1 i j 6 . (Feel free to write a program to do it, if that will be faster). c. Sum up the columns and rows of your chart and state an interesting theorem about the number of rolls whose second highest die value is m. d. A generalization of this theorem for n dice, where n is an odd number, is that the number of rolls of n dice whose median value die equals m is the same as the number of rolls of n dice whose median value die equals 7 - m. Prove this theorem. 2. In the game of Risk, one player, the attacker, throws three dice. The defender chooses to roll either one die or two dice. If the defender rolls two dice, then the two highest rolls of the attacker are compared with the rolls of the defender. The higher is compared against the higher and the lower against the lower, with a tie going to the defender. In every battle with two dice, the defender can win two, lose two or split. If the defender rolls just one die, then only the highest roll of the attacker is compared to it, and again a tie goes to the defender. Here the defender can win one or lose one.
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