Selecting a Team of 5 Players Involving Specific Strongest and Weakest Players

In many scenarios, such as sports teams, tournament brackets, or any collective decision-making process, it is necessary to form a team with specific individuals included. This article discusses how to select a team of 5 players from a group of 10 players, ensuring that both the strongest and weakest players are included. We will use the mathematical principles of combination to determine the number of ways this can be accomplished.

How to Choose a Team of 5 Players Including Both the Strongest and Weakest Players

The process can be broken down into a few distinct steps, allowing for a clear and methodical solution. Let's explore these steps in detail.

Step 1: Including the Strongest and Weakest Players

Since we must include both the strongest and weakest players, we begin by selecting these two individuals. This process can be described as selecting 2 out of 10 players, which is represented as {10 choose 2}.

The formula for combination is given by:

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

Here, n is the total number of items to choose from, and r is the number of items to choose. In this case, we have n 10 and r 2:

(binom{10}{2} frac{10!}{2!8!} frac{10 times 9}{2 times 1} 45)

However, since we are specifically choosing the strongest and the weakest players, the number of combinations is simplified to 1 (there is only one way to choose both individuals).

Step 2: Selecting the Remaining Players

After including the strongest and weakest players, we need to choose the remaining 3 players from the 8 remaining players. We use the combination formula again, with n 8 and r 3:

(binom{8}{3} frac{8!}{3!(8-3)!} frac{8 times 7 times 6}{3 times 2 times 1} frac{336}{6} 56)

Conclusion

In summary, the total number of ways to form a team of 5 players, ensuring that both the strongest and weakest players are included, can be calculated by multiplying the number of ways to choose the remaining 3 players from the 8 remaining players. The calculation is therefore 1 × 56 56 ways.

Advanced Application and Further Reading

The principles discussed here are based on the binomial coefficient, a fundamental concept in combinatorial mathematics. For those interested in further exploring this area, consider delving into related topics such as permutations, multinomial coefficients, and probability theory. Understanding these concepts can provide valuable insights into a wide range of real-world applications, from statistical analysis to algorithm design.