Correcting a Common Misconception in Combination Problems: Selecting Four People from Six Couples

When solving problems involving combinations and permutations, it's crucial to consider the order and the specific requirements. A common pitfall in such problems is neglecting to account for the order of selection, which can lead to overcounting or undercounting. In the context of selecting four people from six couples with the condition that exactly one couple is included, the given logic is almost correct, but it encounters a critical step that needs adjustment. Let's delve into this step-by-step.

Problem Statement

The problem is to select four people from six couples such that exactly one couple is included. This involves selecting one couple from six, then selecting one person from the remaining five couples, and finally, selecting one person from the remaining eight individuals.

Original Logic

Initially, the logic presented is as follows:

Select one couple from six: This can be done in 6C1 ways. Select one couple from the remaining five: This can be done in 5C1 ways. Select one person from the chosen couple (2C1): This can be done in 2C1 ways. Select one person from the remaining eight individuals: This can be done in 8C1 ways.

The total number of ways, according to this logic, is 6C1 × 5C1 × 2C1 × 8C1 480.

Identifying the Error

While the steps are mathematically correct up to a certain point, there is a misunderstanding regarding the order of selection.

After selecting one couple, you are left with five couples or ten individuals to choose from. However, the subsequent selection of one individual from these ten can lead to overcounting. The error lies in the third step, where the selection is counted as if it were an ordered permutation rather than an unordered combination.

To clarify, let's consider a smaller example with two couples: AB and CD. Following the same logic:

Select one couple from two: This can be done in 2C1 ways. Select one person from the chosen couple: This can be done in 2C1 ways. Select one person from the remaining two individuals: This can be done in 2C1 ways.

The total number of ways, according to this logic, would be 2C1 × 2C1 × 2C1 8. However, this overcounts the combinations because AC and CA are considered different, as are AD and DA, when in fact, AC is the same as CA, and AD is the same as DA.

Correct Approach

The correct approach is to recognize that the order of selection does not matter here. Instead of treating it as a permutation, we should treat it as a combination.

Let's count the total number of ways again but in a more systematic manner:

Select one couple from six: This can be done in 6C1 ways. After selecting one couple, you have 10 individuals left, from which one person needs to be chosen: This can be done in 10C1 ways. Finally, one of the remaining 8 individuals needs to be chosen: This can be done in 8C1 ways.

The initial calculation is correct up to the point of 10C1, but the last step requires accounting for the order:

Since the order does not matter, we should divide by 2! to correct for the overcounting:

Total number of ways 6C1 × 5C1 × 2C1 × 10C1 / 2! 6 × 5 × 2 × 10 / 2 300.

This results in the correct answer, 300 ways, which is the revised count.

Conclusion

Solving problems involving combinations and permutations requires careful consideration of the specific conditions and the order of selection. In the case of selecting four people from six couples with exactly one couple included, the initial approach was close but required an adjustment to account for the overcounting caused by treating the order as significant when it is not.

Understanding and correctly applying these principles is essential for solving similar problems and ensuring accurate results in combinatorial analysis.