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List the elements in each of the sets. The set \(F\) consisting of the four seasons.

List the elements in each of the sets. The set of all outcomes of rolling two distinguishable dice such that the numbers add to 8 .

How many different four-letter sequences can be formed from the letters a, a, a, b? HINT [See Example 3.]

Ice Cream At the beginning of 2002, Baskin-Robbins claimed to have "nearly 1,000 different ice cream flavors." \(^{\prime \prime}\) Assuming that you could choose from 1,000 different flavors, that you could have your ice cream in a cone, a cup, or a sundae, and that you could choose from a dozen different toppings, how many different desserts could you have?

Evaluate each number. $$ C(100,97) $$

Let \(S=\\{\) Barnsley, Manchester United, Southend, Sheffield United, Liverpool, Maroka Swallows, Witbank Aces, Roval Tigers, Dundee United, Lyon' be a universal set, \(A=\\{\) Southend, Liverpool, Maroka Swallows, Royal Tigers\\}, and \(B=\\{\) Barnsley, Manchester United, Southend\\}. Find the numbers indicated. \(n\left(A^{\prime}\right)\)

Reward While selecting candy for students in his class, Professor Murphy must choose between gummy candy and licorice nibs. Gummy candy packets come in three sizes, while packets of licorice nibs come in two. If he chooses gummy candy, he must select gummy bears, gummy worms, or gummy dinos. If he chooses licorice nibs, he must choose between red and black. How many choices does he have? HINT [See Example 2.]

How many three-letter (unordered) sets are possible that use the letters \(\mathrm{q}, \mathrm{u}, \mathrm{a}, \mathrm{k}, \mathrm{e}, \mathrm{s}\) at most once each?

Tests A test requires that you answer first Part A and then either Part B or Part C. Part A consists of four true-false questions, Part B consists of four multiple-choice questions with one correct answer out of five, and Part C consists of three questions with one correct answer out of six. How many different completed answer sheets are possible?

Stock Portfolios Your broker has suggested that you diversify your investments by splitting your portfolio among mutual funds, municipal bond funds, stocks, and precious metals. She suggests four good mutual funds, three municipal bond funds, eight stocks, and three precious metals (gold, silver. and platinum). a. Assuming your portfolio is to contain one of each type of investment, how many different portfolios are possible? b. Assuming your portfolio is to contain three mutual funds. two municipal bond funds, one stock, and two precious metals, how many different portfolios are possible?