1. **Problem statement:** Mrs. Rivera has 10 newest gowns and 5 mannequins. She wants to display 5 gowns at a time and change the set every 2 days. We need to find how many days will pass before she runs out of a new set of gowns to display. 2. **Understanding the problem:** Each display uses 5 gowns out of 10. We want to find how many unique sets of 5 gowns can be made from 10 gowns. 3. **Formula used:** The number of ways to choose $k$ items from $n$ items without regard to order is given by the combination formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ 4. **Calculate the number of unique sets:** $$\binom{10}{5} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252$$ 5. **Interpretation:** There are 252 unique sets of gowns that can be displayed. 6. **Calculate total days before running out of new sets:** Since each set is displayed for 2 days, $$252 \text{ sets} \times 2 \text{ days/set} = 504 \text{ days}$$ 7. **Final answer:** Mrs. Rivera will run out of new sets after 504 days.