1. **State the problem:** We want to find how many different sundaes can be made by choosing 3 ice cream flavors out of 31, 3 sauces out of 7, and 3 toppings out of 10. 2. **Formula used:** The number of ways to choose $k$ items from $n$ items without regard to order is given by the combination formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ 3. **Calculate each combination:** - Ice cream flavors: $\binom{31}{3} = \frac{31!}{3!\times 28!} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1} = 4495$ - Sauces: $\binom{7}{3} = \frac{7!}{3!\times 4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$ - Toppings: $\binom{10}{3} = \frac{10!}{3!\times 7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$ 4. **Multiply the combinations:** Since the choices are independent, multiply the number of ways: $$4495 \times 35 \times 120 = 18,891,000$$ 5. **Final answer:** There are **18,891,000** different sundaes possible by choosing 3 ice cream flavors, 3 sauces, and 3 toppings.