Introduction:

In sports and recreational activities, organizing games correctly is crucial for ensuring a fair and enjoyable experience for all participants. For mixed doubles games among eight couples, how many ways can these games be organized if the husband and wife from each couple must always play in the same team? In this article, we will explore the mathematical principles involved in such a scenario.

Understanding the Scenario

The problem at hand involves organizing a mixed doubles game among eight married couples under the condition that each husband and wife must play on the same team. A mixed doubles game typically consists of two teams, each with one man and one woman. This restriction significantly reduces the possible combinations.

Combinatorial Analysis

Given that each husband and wife must be on the same team, let's consider the possible ways to assign players. Since there are 8 married couples, we have 8 possible teams, each consisting of one husband and one wife.

Once we select a couple to start, we are left with 7 other couples to form the second team. The number of ways to form such teams can be calculated using combinations. Specifically, we need to determine how many ways we can select 2 out of the 8 couples to form the teams, as the remaining couples will automatically form the opposing team.

The combinatorial expression is given by the binomial coefficient ( binom{8}{2} ), which represents the number of ways to choose 2 items (or couples) out of 8. The formula for the binomial coefficient is:

( binom{n}{k} frac{n!}{k!(n-k)!} )

Substituting ( n 8 ) and ( k 2 ) gives:

( binom{8}{2} frac{8!}{2!(8-2)!} frac{8 times 7}{2 times 1} 28 )

Therefore, there are 28 possible ways to organize the mixed doubles game among the 8 couples under the given condition.

Conclusion

Organizing a mixed doubles game among eight married couples where each husband and wife must play on the same team can be systematically approached using combinatorial mathematics. The number of possiblebinations is 28, reflecting the various ways to pair the couples into two teams.

Understanding these principles can help in planning and organizing similar events, ensuring fairness and maximizing participant satisfaction.