Question: You order a pizza at a restaurant. You can choose 3 toppings from a list of 10. How many combinations of toppings are possible?
1. **State the problem:** We want to find how many different combinations of 3 toppings can be chosen from 10 available toppings.
2. **Formula used:** The number of combinations of choosing $k$ items from $n$ items is given by the binomial coefficient:
$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
where $n!$ denotes the factorial of $n$.
3. **Apply the formula:** Here, $n=10$ and $k=3$.
$$\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!}$$
4. **Simplify the factorial expression:**
$$\frac{10 \times 9 \times 8 \times \cancel{7!}}{3! \times \cancel{7!}} = \frac{10 \times 9 \times 8}{3!}$$
5. **Calculate $3!$:**
$$3! = 3 \times 2 \times 1 = 6$$
6. **Divide numerator by denominator:**
$$\frac{10 \times 9 \times 8}{6} = \frac{720}{6} = 120$$
7. **Conclusion:** There are $120$ possible combinations of 3 toppings from 10.