1. **Restate the fourth point:** We need to calculate the binomial coefficient $\binom{10}{3}$, which counts how many ways we can choose 3 successes out of 10 trials. 2. **Formula for binomial coefficient:** $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ (n factorial) is the product of all positive integers up to $n$. 3. **Apply the formula:** $$\binom{10}{3} = \frac{10!}{3! \times 7!}$$ 4. **Simplify factorials:** Since $10! = 10 \times 9 \times 8 \times 7!$, the $7!$ cancels out: $$\binom{10}{3} = \frac{10 \times 9 \times 8 \times 7!}{3! \times 7!} = \frac{10 \times 9 \times 8}{3!}$$ 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. **Interpretation:** There are 120 different ways to choose 3 successes out of 10 trials. This step is crucial because it tells us how many different sequences of successes and failures result in exactly 3 successes, which is why it appears in the probability formula.