Show All Work In Horse Racing: A Trifecta Bet Is One Where

Show All Work1 In Horse Racing A Trifecta Bet Is One Wher

In horse racing, a trifecta bet requires selecting the first three finishers of a race in exact order, while a boxed trifecta bet involves selecting the first three finishers in any order. Given 18 horses in the 2015 Kentucky Derby, the problem asks for the number of possible bets:

Paper For Above instruction

Part a): To determine how many different possible trifecta bets could be made when selecting the first three finishers in exact order, we are looking for permutations of 18 horses taken 3 at a time. The formula for permutations is:

P(n, r) = n! / (n - r)!

Where n = 18 and r = 3:

P(18, 3) = 18! / (18 - 3)! = 18! / 15! = 18 × 17 × 16 = 4896

Thus, there are 4,896 different possible trifecta bets in exact order.

Part b): For a boxed trifecta, the order of the first three finishers does not matter. Therefore, we are counting combinations of 18 horses taken 3 at a time:

C(n, r) = n! / [r! (n - r)!]

C(18, 3) = 18! / [3! × 15!] = (18 × 17 × 16) / (3 × 2 × 1) = 816

Therefore, there are 816 possible boxed trifecta bets.

References

  • Ross, S. M. (2014). Introduction to Probability and Statistics. Academic Press.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications. Cengage Learning.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • McClave, J. T., & Sincich, T. (2012). A First Course in Statistics. Pearson.

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