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Consider a tournament of n contestants in which the
Chapter 1, Problem 16TE(choose chapter or problem)
Problem 16TE
Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let N(n) denote the number of different possible outcomes. For instance, N(2) = 3, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first.
(a) List all the possible outcomes when n = 3.
(b) With N(0) defined to equal 1, argue, without any computations, that
(c) Show that the formula of part (b) is equivalent to the following:
(d) Use the recursion to find N(3) and N(4).
Questions & Answers
QUESTION:
Problem 16TE
Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let N(n) denote the number of different possible outcomes. For instance, N(2) = 3, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first.
(a) List all the possible outcomes when n = 3.
(b) With N(0) defined to equal 1, argue, without any computations, that
(c) Show that the formula of part (b) is equivalent to the following:
(d) Use the recursion to find N(3) and N(4).
ANSWER:
Step 1 of 5
Let N (n) denote the number of different possible outcomes. For instance, N (2) = 3, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first.
Here our goal is:
a). We need to list all the possible outcomes when n = 3.
b). When N(0) = 1, We need to argue, without any computations, that:
c). We need to show that the formula of part (b) is equivalent to the following:
d). We need to find N (3) and N (4).