1. I have nine paintings and have room to display only three of them at a time on my wall. How many different ways could I do this?
2. In the Colorado Powerball Lottery, 69 balls numbered 1 through 69 are placed in a machine and five of them are drawn at random. If the five numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order the numbers are drawn in doesn’t matter. Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket.
3. In Europe, the casino game roulette uses a wheel with 37 spaces (18 red, 18 black, and 1 green). In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they lose their $1. What is the expected value of this $1 bet? On average, how much money should a player expect to win or lose per bet if they play this game repeatedly?