There are three boys in a class of nine students. Three students are to be randomly selected (without replacement) to help with bus duty. Determine the probability that all three students are boys. Show how you determined your answer.

To determine the probability that all three students selected are boys, we can follow these steps:

1. Determine the total number of ways to select 3 students out of 9: This can be calculated using the combination formula $\binom{n}{k}$, which represents the number of ways to choose $k$ items from $n$ items without regard to order. $\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3! \cdot 6!}$ Simplifying this: $\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$

2. Determine the number of ways to select 3 boys out of 3: Since there are exactly 3 boys and we need to select all 3 of them, the number of ways to do this is: $\binom{3}{3} = \frac{3!}{3!(3-3)!} = \frac{3!}{3! \cdot 0!} = 1$

3. Calculate the probability: The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. $\text{Probability} = \frac{\text{Number of ways to select 3 boys}}{\text{Total number of ways to select 3 students}} = \frac{1}{84}$

Therefore, the probability that all three students selected are boys is $\frac{1}{84}$.