1. **State the problem:** There are 7 people who check their coats, and all claim checks are lost. The coats are returned randomly. We want to find the probability that each person gets their own coat back. 2. **Formula and explanation:** The total number of ways to return 7 coats to 7 people is $7!$ (7 factorial). 3. The number of ways that everyone gets their own coat back is exactly 1 (the perfect matching). 4. Therefore, the probability is given by: $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{7!}$$ 5. Calculate $7!$: $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$$ 6. So the probability is: $$\frac{1}{5040} \approx 0.000198$$ **Final answer:** The probability that all 7 people get their own coats back is $\frac{1}{5040}$ or approximately 0.000198.