Ten nominees for an award are seated at a round table at a banquet. (a) Three awardees are chosen. How many different sets of three winners are possible? (b) There is actually a ranking among the three awardees, with first, second, and third places winning different prizes. How many different ordered slates of three winners are possible? (c) After the winners are announced, one nominee remarks to another, "Isn't it in- teresting that none of the winners were sitting next to each other?" How many ways are there to pick a set of three winners from the table, none of whom are seated next to one another? For this problem, we don't care who comes in first, second, and third. Hint: Define A; as the set of all outcomes where nominees i and i+ 1 are both chosen. The bad outcomes are then |A₁ U A₂ U... U A10| which seems like too much to ask via PIE, but luckily most intersections are very easy to count. For example, A₁ A₂] = 1 (why?) and A₁ A3] = 0.

Ten nominees for an award are seated at a round table at a banquet. (a) Three awardees are chosen. How many different sets of three winners are possible? (b) There is actually a ranking among the three awardees, with first, second, and third places winning different prizes. How many different ordered slates of three winners are possible? (c) After the winners are announced, one nominee remarks to another, "Isn't it in- teresting that none of the winners were sitting next to each other?" How many ways are there to pick a set of three winners from the table, none of whom are seated next to one another? For this problem, we don't care who comes in first, second, and third. Hint: Define A; as the set of all outcomes where nominees i and i+ 1 are both chosen. The bad outcomes are then |A₁ U A₂ U... U A10| which seems like too much to ask via PIE, but luckily most intersections are very easy to count. For example, A₁ A₂] = 1 (why?) and A₁ A3] = 0.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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