Step 1: Understanding the Concept:
Probability is defined as the ratio of favorable outcomes to the total possible outcomes.
When we select multiple items from a group and the order doesn't matter, we use "Combinations."
In this problem, we have a mixed group of men and women, and we need to calculate the probability of a specific combination (1 man and 2 women).
The total sample space is all possible ways to pick 3 people from the entire group.
Step 2: Key Formula or Approach:
The combination formula is:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
Probability \( P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} \).
Step 3: Detailed Explanation:
Calculate the Total Sample Space (\( n(S) \)):
There are 4 women and 3 men, so total candidates = 7.
We need to choose 3 candidates out of 7:
\[ n(S) = \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
Now, calculate the Favorable Outcomes (\( n(E) \)):
We need exactly 1 man from the 3 available men:
\[ \binom{3}{1} = 3 \]
We need exactly 2 women from the 4 available women:
\[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \]
Using the Multiplication Principle, the total number of ways to pick this specific team is:
\[ n(E) = 3 \times 6 = 18 \]
Finally, determine the probability:
\[ P = \frac{18}{35} \]
Step 4: Final Answer:
The probability of selecting one man and two women is 18/35.