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You are dealt one card from a 52-card deck. Find the probability that you are dealt a 5 or a black card.

Suppose you are asked to list, in order of preference, the three best movies you have seen this year. If you saw 20 movies during the year, in how many ways can the three best be chosen and ranked?

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{5}=\frac{5}{10}+\frac{5}{100}+\frac{5}{1000}+\frac{5}{10,000}+\cdots $$

Find the term indicated in each expansion. $$ \left(x-\frac{1}{2}\right)^{9} ; \text { fourth term } $$

In a race in which six automobiles are entered and there are no ties, in how many ways can the first three finishers come in?

Fermat鈥檚 most notorious theorem, described in the section opener on page 1098, baffled the greatest minds for more than three centuries. In 1994, after ten years of work, Princeton University鈥檚 Andrew Wiles proved Fermat鈥檚 Last Theorem. People magazine put him on its list of 鈥渢he 25 most intriguing people of the year,鈥� the Gap asked him to model jeans, and Barbara Walters chased him for an interview. 鈥淲ho鈥檚 Barbara Walters?鈥� asked the bookish Wiles, who had somehow gone through life without a television. Using the 1993 PBS documentary 鈥淪olving Fermat: Andrew Wiles鈥� or information about Andrew Wiles on the Internet, research and present a group seminar on what Wiles did to prove Fermat鈥檚 Last Theorem, problems along the way, and the role of mathematical induction in the proof.

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$ \sum_{i=1}^{17}(5 i+3) $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 2+2^{2}+2^{3}+\cdots+2^{11} $$

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{1}=\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10,000}+\cdots $$

In a production of West Side Story, eight actors are considered for the male roles of Tony, Riff, and Bernardo. In how many ways can the director cast the male roles?