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While on a Saturday shopping spree Jennifer and Tiffany witnessed two men driving away from the front of a jewelry shop, just before a burglar alarm started to sound. Although everything happened rather quickly, when the two young ladies were questioned they were able to give the police the following information about the license plate (which consisted of two letters followed by four digits) on the get-away car. Tiffany was sure that the second letter on the plate was either an \(\mathrm{O}\) or \(\mathrm{a} \mathrm{Q}\) and the last digit was either a 3 or an 8 . Jennifer told the investigator that the first letter on the plate was either a \(\mathrm{C}\) or a \(\mathrm{G}\) and that the first digit was definitely a 7. How many different license plates will the police have to check out?

a) In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar, and five (identical) Susan B. Anthony dollars? b) In how many ways can we select \(n\) objects from a collection of size \(2 n\) that consists of \(n\) distinct and \(n\) identical objects?

To raise money for a new municipal pool, the chamber of commerce in a certain city sponsors a race. Each participant pays a \(\$ 5\) entrance fee and has a chance to win one of the differentsized trophies that are to be awarded to the first eight runners who finish. a) If 30 people enter the race, in how many ways will it be possible to award the trophies? b) If Roberta and Candice are two participants in the race, in how many ways can the trophies be awarded with these two runners among the top three?

If \(n\) is a positive integer and \(n>1\), prove that \(\left(\begin{array}{c}n \\\ 2\end{array}\right)+\left(\begin{array}{c}n-1 \\ 2\end{array}\right)\) is a perfect square. 7\. A committee of 12 is to be selected from 10 men and 10 women. In how many ways can the selection be carried out if (a) there are no restrictions? (b) there must be six men and six women? (c) there must be an even number of women? (d) there must be more women than men? (e) there must be at least eight men?

A certain "Burger Joint" advertises that a customer can have his or her hamburger with or without any or all of the following: catsup, mustard, mayonnaise, lettuce, tomato, onion, pickle, cheese, or mushrooms. How many different kinds of hamburger orders are possible?

In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels?

In how many ways can a gambler draw five cards from a standard deck and get (a) a flush (five cards of the same suit)? (b) four aces? (c) four of a kind? (d) three aces and two jacks? (e) three aces and a pair? (f) a full house (three of a kind and a. pair)? (g) three of a kind? (h) two pairs?

Patter's Pastry Parlor offers eight different kinds of pastry and six different kinds of muffins. In addition to bakery items one can purchase small, medium, or large containers of the following beverages: coffee (black, with cream, with sugar, or with cream and sugar), tea (plain, with cream, with sugar, with cream and sugar, with lemon, or with lemon and sugar), hot cocoa, and orange juice. When Carol comes to Patter's, in how many ways can she order a) one bakery item and one medium-sized beverage for herself? b) one bakery item and one container of coffee for herself and one muffin and one container of tea for her boss, Ms. Didio? c) one piece of pastry and one container of tea for herself, one muffin and a container of orange juice for Ms. Didio, and one bakery item and one container of coffee for each of her two assistants, Mr. Talbot and Mrs. Gillis?

Pamela has 15 different books. In how many ways can she place her books on two shelves so that there is at least one book on each shelf? (Consider the books in each arrangement to be stacked one next to the other, with the first book on each shelf at the left of the shelf.)

How many ways are there to pick a five-person basketball team from 12 possible players? How many selections include the weakest and the strongest players?