Selecting 8 People from 6 Couples with at Least 3 Couples

Combinatorial problems often challenge our understanding of permutations and combinations. In this article, we explore a specific scenario: selecting 8 people from 6 couples such that there are at least 3 couples. We break down the problem into manageable cases and apply combinatorial principles to find the solution.

Problem Statement

Given 6 couples, we want to determine how many ways we can select 8 people such that there are at least 3 couples among the selected individuals. Each couple consists of 2 individuals, and the selection criteria must be met to ensure that the problem is well-defined.

Case Analysis

Case 1: 3 Couples

In this case, we choose 3 couples from the 6 couples available. The number of ways to do this is given by the binomial coefficient binom{6}{3}. After selecting these 3 couples, we have 6 people, and we need to select 2 more people from the remaining 3 couples. We can select 1 person from each of the remaining 2 couples, giving us binom{3}{2} ways to choose the couples, and 2 choices for each couple. Thus, the total number of ways for this case is:

binom{6}{3} times binom{3}{2} times 2^2

Case 2: 4 Couples

In this case, we select 4 couples from the 6. Since we are selecting 4 couples, we automatically select 8 people, and no further selection is needed. The number of ways to do this is given by:

binom{6}{4}

Case 3: 5 Couples

In this case, we choose 5 couples from the 6. We select 1 person from each of the 5 couples, leaving us with 5 people. We then select 3 more people from the remaining 1 couple. The number of ways to do this is given by:

binom{6}{5} times 2^1

Case 4: 6 Couples

In this case, we choose all 6 couples, resulting in 6 people. We then select 2 more people from the remaining couples, which is not necessary as we already have 8 people. The number of ways to do this is given by:

binom{6}{6} times 2^2

Final Calculation

Now, we sum the totals from all cases:

text{Total} 240 15 12 4 271

Therefore, the total number of ways to select 8 people such that there are at least 3 couples is 271.

Conclusion

This problem showcases the application of combinatorial principles in solving selection problems. By breaking down the scenario into individual cases and applying the appropriate binomial coefficients, we can systematically determine the total number of valid selections.

Keywords: combinatorics, selection problems, couples combinations