1. **Problem statement:** You roll a fair 6-sided die 17 times. Find the probability of getting exactly 5 even numbers and exactly 3 fives. 2. **Understanding the problem:** Each roll can result in one of three mutually exclusive outcomes relevant to the problem: - Even number (2, 4, or 6) - Number 5 - Odd number other than 5 (1 or 3) 3. **Total trials:** 17 rolls. 4. **Events to count:** Exactly 5 even numbers, exactly 3 fives, and the remaining 9 rolls must be odd numbers other than 5. 5. **Probabilities for each category per roll:** - Probability(even) = 3/6 = 1/2 - Probability(5) = 1/6 - Probability(odd other than 5) = 2/6 = 1/3 6. **Using multinomial probability formula:** $$P = \frac{17!}{5!3!9!} \times \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{6}\right)^3 \times \left(\frac{1}{3}\right)^9$$ 7. **Calculate factorial term:** $$\frac{17!}{5!3!9!} = 6188$$ 8. **Calculate probability term:** $$\left(\frac{1}{2}\right)^5 = \frac{1}{32}$$ $$\left(\frac{1}{6}\right)^3 = \frac{1}{216}$$ $$\left(\frac{1}{3}\right)^9 = \frac{1}{19683}$$ 9. **Multiply all terms:** $$P = 6188 \times \frac{1}{32} \times \frac{1}{216} \times \frac{1}{19683}$$ 10. **Simplify:** $$P = \frac{6188}{32 \times 216 \times 19683} = \frac{6188}{136048896} \approx 0.0000455$$ 11. **Final answer:** The probability is approximately **0.0000** when rounded to 4 decimal places. **Note:** The given answer 0.0100 does not match the exact multinomial calculation; the correct probability is much smaller.