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In the California Daily 3 game, a contestant must select three numbers among 0 to \(9 .\) One type of "box play" win requires that three numbers match in any order those randomly drawn by a lottery representative, repetitions allowed. What is the probability of choosing the winning numbers, assuming that the contestant chooses three distinct numbers?

Answer to give an argument that proves the following result. A sequence \(a_{1}, a_{2}, \ldots, a_{n^{2}+1}\) of \(n^{2}+1\) distinct numbers contains either an increasing subsequence of length \(n+1\) or a decreasing subsequence of length \(n+1 .\) Suppose by way of contradiction that every increasing or decreasing subsequence has length \(n\) or less. Let \(b_{i}\) be the length of a longest increasing subsequence starting at \(a_{i},\) and let \(c_{i}\) be the length of a longest decreasing subsequence starting at \(a_{i} .\) Explain why \(1 \leq b_{i} \leq n\) and \(1 \leq c_{i} \leq n\).

Answer to give an argument that proves the following result. A sequence \(a_{1}, a_{2}, \ldots, a_{n^{2}+1}\) of \(n^{2}+1\) distinct numbers contains either an increasing subsequence of length \(n+1\) or a decreasing subsequence of length \(n+1 .\) Suppose by way of contradiction that every increasing or decreasing subsequence has length \(n\) or less. Let \(b_{i}\) be the length of a longest increasing subsequence starting at \(a_{i},\) and let \(c_{i}\) be the length of a longest decreasing subsequence starting at \(a_{i} .\) What is the contradiction?

Find the probability of obtaining a bridge hand with \(5-4-2-2\) distribution, that is, five cards in one suit, four cards in another suit, and two cards in each of the other two suits.

Find the number of integer solutions of $$x_{1}+x_{2}+x_{3}=15$$ \(x_{1} \geq 0, x_{2}>0, x_{3}=1\)

Two dice are rolled, one blue and one red. How many outcomes have exactly one die showing \(2 ?\)

Two dice are rolled, one blue and one red. How many outcomes have at least one die showing \(2 ?\)

Suppose that a pizza parlor features four specialty pizzas and pizzas with three or fewer unique toppings (no choosing anchovies twice!) chosen from 17 available toppings. How many different pizzas are there?

Twelve basketball players, whose uniforms are numbered 1 through \(12,\) stand around the center ring on the court in an arbitrary arrangement. Show that some three consecutive players have the sum of their numbers at least 20 .

Two dice are rolled, one blue and one red. How many outcomes give an even sum?