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Find the number of (unordered) five-card poker hands, selected from an ordinary 52 -card deck, having the properties indicated. Containing cards of all suits

Exercises \(47-50\) ask about the following situation. In a small charity fundraiser, 70 tickets are sold numbered 1 through \(70 .\) Each person buys one ticket. Later in the evening. 20 numbers are randomly drawn from among \(I\) through 70 , and those holding these numbers win modest prizes. Among those buying the tickets are Maya and Chloe. What is the probability that both Maya and Chloe win prizes?

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer: How many selections are there in which Ben is either chairperson or treasurer?

Find the least \(n\) such that among \(n\) persons, the probability that at least two persons have birthdays on April 1 (but not necessarily in the same year) is greater than \(1 / 2 .\) Assume that all months and dates are equally likely, and ignore February 29 birthdays.

A two-person game is played in which a fair coin is tossed until either the sequence HT (heads, tails) or the sequence TT (tails, tails) appears. If HT appears, the first player wins; if TT appears, the second player wins. Would you rather be the first or second player? Explain.

Refer to the integers from 5 to 200 , inclusive. How many numbers are there?

Refer to the integers from 5 to 200 , inclusive. How many consist of distinct digits?

Refer to the integers from 5 to 200 , inclusive. How many do not contain the digit \(0 ?\)

Refer to the integers from 5 to 200 , inclusive. How many are greater than 101 and do not contain the digit \(6 ?\)

Refer to the integers from 5 to 200 , inclusive. How many have the digits in strictly increasing order? (Examples are \(13,147,8 .)\)