91Ó°ÊÓ

Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in \( 30 \) states, Washington D.C., and the U.S. Virgin Islands.The game is played by drawing five white balls out of a drum of \( 59 \) white balls (numbered \( 1 - 59 \)) and one red powerball out of a drum of \( 39 \) red balls (numbered \( 1 - 39 \)). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers if the jackpot is won by matching all five white balls in order and the red power ball. (c) Compare the results of part (a) with a state lottery in which a jackpot is won by matching six balls from a drum of \( 59 \) balls.

In Exercises 79 - 82, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion \( \left(1.02\right)^8 = \left(1 + 0.02\right)^8 = 1 + 8\left(0.02\right) + 28\left(0.02\right)^2 + \cdots \). \( \left(2.99\right)^{12} \)

In Exercises 85-96, find the sum. \( \displaystyle \sum_{k=1}^{4} 10 \)

In Exercises 85 - 88, consider independent trials of an experiment in which each trial has two possible outcomes: success or failure. The probability of a success on each trial is \( p \), and the probability of a failure is \( q = 1 - p \).In this context, the term \(_nC_kp^kq^{n - k} \) in the expansion of \( \left(p + q\right)^n \) gives the probability of \( k \) successes in the \( n \) trials of the experiment. The probability of a sales representative making a sale with any one customer is \( \dfrac{1}{3} \) The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term \( _8C_4 \left(\dfrac{1}{3}\right)^4\left(\dfrac{2}{3}\right)^4 \) in the expansion of \( \left(\dfrac{1}{3} + \dfrac{2}{3}\right)^8 \).

Determine the seating capacity of an auditorium with 30 rows of seats if there are 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on.

An object with negligible air resistance is dropped from a plane. During the first second of fall, the object falls \( 4.9 \) meters; during the second second, it falls \( 14.7 \) meters; during the third second, it falls \( 24.5 \) meters; during the fourth second, it falls \( 34.3 \) meters. If this arithmetic pattern continues,how many meters will the object fall in \( 10 \) seconds?

Decide whether each scenario should be counted using permutations or combinations.Explain your reasoning. (Do not calculate.) (a) Number of ways \( 10 \) people can line up in a row for concert tickets. (b) Number of different arrangements of three types of flowers from an array of \( 20 \) types. (c) Number of four-digit pin numbers for a debit card. (d) Number of two-scoop ice cream sundaes created from \( 31 \) different flavors.

A county fair is holding a baked goods competition in which the top eight bakers receive cash prizes. First place receives a cash prize of \( \$200 \),second place receives \( \$175 \), third place receives \( \$150 \),and so on. (a) Write a sequence that represents the cash prize awarded in terms of the place in which the baked good places. (b) Find the total amount of prize money awarded at the competition.

Form rows \( 8 - 10 \) of Pascals Triangle.

In Exercises 93 - 106, find the sum of the infinite geometric series. \( \sum_{n=0}^{\infty}-10\left(0.2\right)^n \)