The Number Of Ways In Which 10 Candidates A 1 , A 2 , . . . , A 10 A_1,A_2,...,A_{10} A 1 , A 2 , ... , A 10 Can Be Ranked So That A 1 A_1 A 1 Is Always Above A 2 A_2 A 2 , Is:

May 22, 2025 by ADMIN 187 views

**The Number of Ways to Rank 10 Candidates with A1 Always Above A2**

Introduction

In this article, we will explore the problem of ranking 10 candidates, denoted as $A_1, A_2, ..., A_{10}$ , in such a way that $A_1$ is always above $A_2$ . This problem falls under the category of combinatorics, permutations, and combinations. We will delve into the solution by considering the candidates as single objects and then breaking them down into individual candidates.

Understanding the Problem

The problem requires us to find the number of ways to rank the 10 candidates, with the condition that $A_1$ is always above $A_2$ . This means that in any valid ranking, $A_1$ must occupy a higher position than $A_2$ . We need to consider all possible permutations of the candidates while satisfying this condition.

Approach

To solve this problem, we can start by considering $A_1$ and $A_2$ as a single object, denoted as $A_1A_2$ . This allows us to treat the remaining 8 candidates as individual objects. We can then calculate the number of ways to arrange these 9 objects (8 individual candidates + 1 combined object $A_1A_2$ ).

Calculating the Number of Arrangements

We can calculate the number of arrangements of the 9 objects using the formula for permutations:

P(n, r) = \frac{n!}{(n-r)!}

where $n$ is the total number of objects and $r$ is the number of objects being arranged.

In this case, we have 9 objects, so $n = 9$ . We want to arrange all 9 objects, so $r = 9$ . Plugging these values into the formula, we get:

P(9, 9) = \frac{9!}{(9-9)!} = \frac{9!}{0!} = 9!

The number of arrangements of the 9 objects is therefore $9! = 362,880$ .

Accounting for the Condition

However, we need to account for the condition that $A_1$ is always above $A_2$ . This means that for every valid arrangement of the 9 objects, we need to consider the number of ways to arrange $A_1$ and $A_2$ within the combined object $A_1A_2$ .

Since $A_1$ must be above $A_2$ , there are only 2 possible arrangements of $A_1$ and $A_2$ within the combined object: $A_1A_2$ or $A_2A_1$ . Therefore, for every valid arrangement of the 9 objects, there are 2 possible arrangements of $A_1$ and $A_2$ .

Calculating the Final Answer

To calculate the final answer, we need to multiply the number of arrangements of the 9 objects by the number of arrangements of $A_1$ and $A_2$ within the combined object:

\text{Final Answer} = 9! \times 2 = 362,880 \times 2 = 725,760

Therefore, the number of ways to rank the 10 candidates with $A_1$ always above $A_2$ is 725,760.

Conclusion

In this article, we explored the problem of ranking 10 candidates with $A_1$ always above $A_2$ . We used the approach of considering $A_1$ and $A_2$ as a single object and then breaking them down into individual candidates. We calculated the number of arrangements of the 9 objects and accounted for the condition that $A_1$ is always above $A_2$ . The final answer is 725,760.

Additional Information

The problem can be generalized to any number of candidates, not just 10.
The condition that $A_1$ is always above $A_2$ can be replaced with any other condition, such as $A_1$ being above $A_3$ or $A_1$ being below $A_2$ .
The problem can be solved using other methods, such as using the concept of permutations with restrictions.

References

[1] "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
[2] "Permutations and Combinations" by Khan Academy

Q: What is the problem of ranking 10 candidates with A1 always above A2?

A: The problem requires us to find the number of ways to rank the 10 candidates, with the condition that A1 is always above A2. This means that in any valid ranking, A1 must occupy a higher position than A2.

Q: How do we approach this problem?

A: We can start by considering A1 and A2 as a single object, denoted as A1A2. This allows us to treat the remaining 8 candidates as individual objects. We can then calculate the number of ways to arrange these 9 objects (8 individual candidates + 1 combined object A1A2).

Q: What is the formula for calculating the number of arrangements?

A: The formula for calculating the number of arrangements is:

P(n, r) = n! / (n-r)!

where n is the total number of objects and r is the number of objects being arranged.

Q: How do we account for the condition that A1 is always above A2?

A: We need to consider the number of ways to arrange A1 and A2 within the combined object A1A2. Since A1 must be above A2, there are only 2 possible arrangements of A1 and A2 within the combined object: A1A2 or A2A1.

Q: What is the final answer to the problem?

A: The final answer is 725,760, which is the product of the number of arrangements of the 9 objects and the number of arrangements of A1 and A2 within the combined object.

Q: Can this problem be generalized to any number of candidates?

A: Yes, the problem can be generalized to any number of candidates. The condition that A1 is always above A2 can be replaced with any other condition, such as A1 being above A3 or A1 being below A2.

Q: What are some related problems?

A: Some related problems include:

Ranking 10 candidates with A1 always above A3
Ranking 10 candidates with A1 always below A2
Ranking 10 candidates with A1 always above A2 and A3 always below A4

Q: What are some references for this problem?

A: Some references for this problem include:

"Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
"Permutations and Combinations" by Khan Academy

Q: What are some common misconceptions about this problem?

A: Some common misconceptions about this problem include:

Assuming that the number of arrangements is simply 10! (10 factorial)
Failing to account for the condition that A1 is always above A2
Not considering the number of ways to arrange A1 and A2 within the combined object A1A2

Q: How can I apply this problem to real-world scenarios?

A: This problem can be applied to real-world scenarios such as:

Ranking candidates for a job or election
Scheduling events or appointments
Organizing a list of items in a specific order

By understanding the concepts and techniques used to solve this problem, you can apply them to a wide range of real-world scenarios.