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Choosing Officers From a pool of 12 candidates, the offices of president, vice-president, secretary, and treasurer will be filled. In how many different ways can the offices be filled?

Batting Order A baseball coach is creating a nine-player batting order by selecting from a team of 15 players. How many different batting orders are possible?

The sample spaces are large and you should use the counting principles discussed in Section 9.6. ATM personal identification number (PIN) codes typically consist of four-digit sequences of numbers. Find the probability that if you forget your PIN, then you can guess the correct sequence (a) at random and (b) when you recall the first two digits.

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{j=1}^{10}\left(3-\frac{1}{2} j+\frac{1}{2} j^{2}\right)$$

Lottery Choices In the Massachusetts Mass Cash game, a player randomly chooses five distinct numbers from 1 to \(35 .\) In how many ways can a player select the five numbers?

American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered \(1-36,\) of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on \(\quad\) three consecutive spins.

Simplifying a Factorial Expression In Exercises \(63-66,\) simplify the factorial expression. $$ \frac{4 !}{6 !} $$

You and a friend agree to meet at your favorite fast-food restaurant between \(5 : 00\) P.M. and \(6 : 00\) P.M. The one who arrives first will wait 15 minutes for the other, and then will leave (see figure). What is the probability that the two of you will actually meet, assuming that your arrival times are random within the hour?

You drop a coin of diameter \(d\) onto a paper that contains a grid of squares \(d\) units on a side (a) Find the probability that the coin covers a vertex of one of the squares on the grid. (b) Perform the experiment 100 times and use the results to approximate \(\pi .\)

Simplifying a Factorial Expression In Exercises \(63-66,\) simplify the factorial expression. $$ \frac{(n+1) !}{n !} $$