Question

4. A “superfecta” is correctly guessing the top four horses in order in a horse race.

   1. (a) If there are 8 horses in a race, how many different superfectas are possible?

   2. (b) “Boxing” a superfecta means that you get to choose the four horses for the superfecta without regard to order. How many box superfectas are possible?

   3. (c) If a bet on a superfecta costs 10 cents. How much would you expect for a box superfecta to cost? (Hint: Think about how many bets you are implicitly allowed to make by boxing!)

5. Consider the digits {2, 3, 4, 5, 6, 7}. If 4 digits are chosen randomly and without replacement to make a 4-digit number, how many ways can a number larger than 5400 be made?

6. 3 red balls, 4 white balls, and 5 blue balls are mixed up in an urn. 3 are drawn at random.

   (a) How many ways can one of each color be drawn?
   (b) How many ways can three of the same color be drawn?

Answer 1

1)

(a) Number of different superfectas possible = 8*7*6*5 = 1680

(b) Number of box superfectas possible = 8C4 = 70

(c) Expected cost for a box superfecta = 10*24 = 240 cents

2)

Number of ways a number larger than 5400 can be made are

(2*6*5*4 + 1*2*5*4) = 280

Since thousands place can have digit 6 or 7 in which other three digits can be any of the remaining digits of thousands place can have digit 5 in which hundreds place can have 6 or 7

3)

(a) The required number of ways = 3*4*5 = 60

(b) The required number of ways = 3C3 + 4C3 + 5C3 = 15