Calculate the probability that randomly selecting three people from a group of six men and three women were result in selecting three women

To calculate the probability of randomly selecting three women from a group consisting of six men and three women, we need to use **combinatorics**. 

We approach this problem step-by-step:

1. **Total number of people**: 6 men + 3 women = 9 people.
2. **Favorable outcomes**: We need to select all 3 women from the group of 3 women.
   - The number of ways to select 3 women from 3 women is \( \binom{3}{3} = 1 \).
3. **Total possible outcomes**: The number of ways to select 3 people from 9 people is \( \binom{9}{3} \).

Now, calculate \( \binom{9}{3} \):

\[
\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
\]

Thus, the total number of ways to select 3 people from 9 is 84.

4. **Probability**: The probability is the ratio of favorable outcomes to the total outcomes.

\[
\text{Probability} = \frac{\binom{3}{3}}{\binom{9}{3}} = \frac{1}{84}
\]

Thus, the probability of selecting 3 women is \( \frac{1}{84} \).

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Would you like more details or clarifications on any step?

Here are 5 questions that expand on this problem:
1. How would the probability change if there were 4 women instead of 3?
2. How is the combination formula \( \binom{n}{r} \) derived?
3. What is the probability of selecting 2 men and 1 woman?
4. How do combinations differ from permutations in this context?
5. What is the probability of selecting at least 1 woman when selecting 3 people?

**Tip**: In probability, combinations are used when the order of selection doesn't matter, whereas permutations are used when order matters.

Calculate the probability that randomly selecting three people from a group of six men and three women will result in selecting three women.

Learn how to calculate the probability of randomly selecting 3 women from a group of 6 men and 3 women using combinatorics and probability theory. Follow a detailed step-by-step solution and understand the combination formula \( \binom{n}{r} \) used in this calculation. Suitable for students in Grades 9-12.

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