A study contains 30 women and 21 men. If 3 subjects are randomly selected without replacement, what is the probability that they are all men? Round answer to four decimal places.

To find the probability that all 3 selected subjects are men, we need to calculate the following:

1. The total number of people in the study is \(30\) women + \(21\) men = \(51\) people.
2. We are selecting 3 people without replacement, so the total number of ways to select 3 people from 51 is calculated using combinations:
\[
\binom{51}{3} = \frac{51 \times 50 \times 49}{3 \times 2 \times 1} = 20,825
\]
3. Next, we calculate the number of ways to select 3 men from the 21 available men:
\[
\binom{21}{3} = \frac{21 \times 20 \times 19}{3 \times 2 \times 1} = 1,330
\]
4. Now, the probability that all 3 selected subjects are men is the ratio of favorable outcomes to the total outcomes:
\[
P(\text{all men}) = \frac{\binom{21}{3}}{\binom{51}{3}} = \frac{1,330}{20,825} \approx 0.0639
\]

Thus, the probability that all 3 subjects are men is approximately **0.0639**.

Would you like more details or have any questions about this process?

### Related questions:
1. What is the probability of selecting all women instead?
2. What is the probability of selecting at least one man?
3. How does the probability change if the selection is with replacement?
4. What is the probability of selecting exactly two men and one woman?
5. How many different combinations of two men and one woman can be made?

### Tip:
When calculating probabilities involving "without replacement," combinations are typically used, as the order of selection does not matter.

Explore the probability of randomly selecting 3 men from a study containing 30 women and 21 men. Learn how to calculate the probability using combinatorics and the probability formula in this detailed step-by-step guide. Suitable for high school students in Grades 9-12.

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