How many ways are there to choose 3 items from 10?

Problem: How many ways are there to choose 3 items from 10, represented as $\binom{10}{3}$?

Steps

1. Apply the combinations formula
   $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
2. Substitute $n = 10$ and $k = 3$
   $$\frac{10!}{3!(10-3)!}$$
3. Simplify the factorial expression
   $$\frac{10!}{3! \cdot 7!}$$
4. Expand the numerator and denominator
   $$\frac{10 \times 9 \times 8 \times 7!}{3 \times 2 \times 1 \times 7!}$$
5. Cancel the common term $7!$
   $$\frac{10 \times 9 \times 8}{3 \times 2 \times 1}$$
6. Calculate the final product
   $$\frac{720}{6}$$
7. Divide
   $$120$$