Combinations of People Choosing from Different Couples
Introduction
In this article, we explore the mathematical concepts of combinations when selecting individuals from different couples. We delve into the problem of choosing 6 individuals from 10 couples under the condition that they are from different couples and provide multiple methods to arrive at the solution. This is a classic problem in combinatorial mathematics that can be approached through various methodologies, offering valuable insights into efficient problem-solving techniques.
Problem Statement
The problem at hand is to determine the number of ways to select 6 individuals from 10 couples so that no two individuals are from the same couple. We can solve this problem using two primary methods: direct combinatorial methods and a step-by-step selection process.
Method 1: Using Combinatorial Mathematics
The first method involves a direct application of combinatorial mathematics. We start by selecting 6 couples out of the 10, and from each of these couples, we choose one individual. Each of the 6 couples has 2 choices (either of the two individuals). Therefore, the total number of combinations can be calculated as:
Equation 1: 10C6 times; 2^6
10C6 represents the number of ways to choose 6 couples from 10, which is 210. 2^6 represents the number of ways to choose one individual from each of these 6 couples, which is 64. Therefore:
Total Combinations 210 times; 64 13,440.
Method 2: Alternative Approach Using Step-by-Step Selection
This method involves a step-by-step selection process, ensuring that each selection eliminates the partner of the chosen individual from further selections.
Step 1: First Draw
There are 20 individuals in total (10 couples). The first draw offers 20 choices. The second individual must not be the partner of the first, which means one of the 18 remaining individuals can be chosen.
Step 2: Second Draw
The second draw offers 18 choices. Again, the next individual must not be the partner of the already chosen individual, leaving 16 choices for the third draw.
And so on until the sixth draw, which leaves 10 choices. The total number of ways to perform these selections is given by the product of the choices at each step:
Total Combinations 20 times; 18 times; 16 times; 14 times; 12 times; 10 13,440.
Conclusion
The problem of selecting 6 individuals from 10 couples, ensuring none of them are from the same couple, can be solved in 13,440 different ways, as demonstrated by both combinatorial mathematics and a step-by-step selection process. These methods highlight the versatility of combinatorial mathematics in solving real-world problems and the importance of understanding different approaches to a single problem.