Introduction / Context:
This question involves forming a simple couple from two groups, one of men and one of women. A couple is defined here as one man together with one woman. We need to count how many different such couples can be formed.
Given Data / Assumptions:
- Number of men = 6.
- Number of women = 9.
- Each couple consists of exactly one man and one woman.
- All men and women are distinct individuals.
Concept / Approach:
To form a couple, we independently choose one man from the 6 available and one woman from the 9 available. The total number of distinct couples is the product of the number of choices for the man and the number of choices for the woman.
Step-by-Step Solution:
Step 1: Choose a man from the 6 men. There are 6 possible choices.
Step 2: For each fixed choice of man, choose a woman from the 9 women. There are 9 choices for the woman.
Step 3: Use the multiplication rule of counting because the choice of man and choice of woman are independent decisions.
Step 4: Total number of different couples = 6 * 9 = 54.
Verification / Alternative check:
You can think of listing pairs as (man, woman). For each of the 6 men, there is a row of 9 possible women, forming a grid of 6 by 9 = 54 unique ordered pairs. Each pair corresponds to a distinct couple, so the total is 54.
Why Other Options Are Wrong:
26, 52 and 28 do not match the straightforward product 6 * 9. For example, 52 is close to 54 but indicates an incorrect subtraction or miscount of a small number of pairs.
Common Pitfalls:
A frequent error is to accidentally count only one side, such as just the men or just the women, or to use combinations in a way that mixes the two groups incorrectly. Another issue is confusing this simple counting with more complex pairings where multiple couples are chosen at once.
Final Answer:
The number of distinct man woman couples that can be formed from 6 men and 9 women is
54.
Discussion & Comments