55. Eight runners have equal chance of winning the race. What is the probability that a person wins the bet if he selects top 3 runners in the correct order? \( \begin{array}{llll}\text { a) } 5.61 \% & \text { b) } \frac{1}{336} & \text { c) } \frac{1}{112} & \text { d) None of these }\end{array} \)

To determine the probability that a person correctly selects the top 3 runners in the exact order out of 8 runners, we can follow these steps: 1. **Total Possible Outcomes:** - The number of ways to arrange 8 runners in the top 3 positions is given by the permutation formula: \[ P(n, k) = \frac{n!}{(n - k)!} \] where \( n = 8 \) (total runners) and \( k = 3 \) (positions). - Calculating: \[ P(8, 3) = \frac{8!}{(8 - 3)!} = \frac{8!}{5!} = 8 \times 7 \times 6 = 336 \] - So, there are **336** possible ordered outcomes for the top 3 positions. 2. **Favorable Outcome:** - There is **only 1** specific ordered combination that matches the bettor's selection exactly. 3. **Probability Calculation:** \[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{1}{336} \] - Converting to a decimal: \[ \frac{1}{336} \approx 0.002976 \text{ or } 0.2976\% \] 4. **Comparing with Given Options:** - **Option a)** 5.61% — Incorrect. - **Option b)** \(\frac{1}{336}\) — **Correct**. - **Option c)** \(\frac{1}{112}\) — Incorrect. - **Option d)** None of these — Incorrect. **Answer:** **b) \(\frac{1}{336}\)**