There Are 170 Athletes At A Meeting. They Each Shake Hands With Everyone Else. How Many Handshakes Were There?A. Permutation; 7,182 B. None Of These C. Permutation; 86,190 D. Combination; 43,095 E. Combination; 14,365
Introduction
Imagine a scenario where 170 athletes gather at a meeting and each one shakes hands with everyone else. The question arises: how many handshakes take place in total? This problem may seem simple at first, but it requires a deeper understanding of mathematical concepts, particularly permutations and combinations. In this article, we will delve into the world of mathematics to find the correct answer.
Understanding Permutations and Combinations
Before we dive into the problem, let's clarify the difference between permutations and combinations. A permutation is an arrangement of objects in a specific order, whereas a combination is a selection of objects without considering the order. In the context of handshakes, permutations would imply that the order of the handshake matters (e.g., A shakes hands with B, and then B shakes hands with C), whereas combinations would imply that the order does not matter (e.g., A shakes hands with B, and C shakes hands with D).
Calculating Handshakes using Permutations
Let's assume that the order of the handshake matters, and we want to calculate the number of permutations. In this case, each athlete will shake hands with every other athlete, resulting in a total of 169 handshakes for each athlete (since they cannot shake hands with themselves). However, this approach is incorrect because it counts each handshake twice (once for each athlete involved).
To calculate the correct number of permutations, we need to use the formula for permutations:
nPr = n! / (n - r)!
where n is the total number of athletes (170), r is the number of athletes involved in each handshake (2), and ! denotes the factorial function.
Plugging in the values, we get:
170P2 = 170! / (170 - 2)! = 170! / 168! = (170 × 169) / 2 = 14,365
However, this is not the correct answer. We need to consider that each handshake is counted twice in the permutation formula. To account for this, we need to divide the result by 2.
14,365 / 2 = 7,182.5
Since we cannot have a fraction of a handshake, we round down to the nearest whole number:
7,182
Calculating Handshakes using Combinations
Now, let's assume that the order of the handshake does not matter, and we want to calculate the number of combinations. In this case, we can use the formula for combinations:
nCr = n! / (r! × (n - r)!)
where n is the total number of athletes (170), r is the number of athletes involved in each handshake (2), and ! denotes the factorial function.
Plugging in the values, we get:
170C2 = 170! / (2! × (170 - 2)!) = 170! / (2! × 168!) = (170 × 169) / 2 = 14,365
This is the correct answer. Each combination represents a unique set of two athletes shaking hands, and there are 14,365 such combinations.
Conclusion
In conclusion, the correct answer to the problem is 14,365 handshakes. This can be calculated using the formula for combinations, which takes into account the fact that the order of the handshake does not matter. The permutation formula, on the other hand, counts each handshake twice and requires division by 2 to obtain the correct result.
Final Answer
Introduction
In our previous article, we explored the mathematical concept of handshakes and how to calculate the number of handshakes that take place in a group of athletes. We also discussed the difference between permutations and combinations and how to use the formulas for each to find the correct answer. In this article, we will continue to delve into the world of mathematics and answer some frequently asked questions related to handshakes and permutations.
Q&A
Q: What is the difference between permutations and combinations?
A: Permutations are arrangements of objects in a specific order, whereas combinations are selections of objects without considering the order. In the context of handshakes, permutations would imply that the order of the handshake matters (e.g., A shakes hands with B, and then B shakes hands with C), whereas combinations would imply that the order does not matter (e.g., A shakes hands with B, and C shakes hands with D).
Q: How do I calculate the number of permutations?
A: To calculate the number of permutations, you can use the formula:
nPr = n! / (n - r)!
where n is the total number of athletes (170), r is the number of athletes involved in each handshake (2), and ! denotes the factorial function.
Q: Why do I need to divide the result by 2 when calculating permutations?
A: When calculating permutations, each handshake is counted twice (once for each athlete involved). To account for this, you need to divide the result by 2.
Q: How do I calculate the number of combinations?
A: To calculate the number of combinations, you can use the formula:
nCr = n! / (r! × (n - r)!)
where n is the total number of athletes (170), r is the number of athletes involved in each handshake (2), and ! denotes the factorial function.
Q: What is the correct answer to the problem?
A: The correct answer to the problem is 14,365 handshakes. This can be calculated using the formula for combinations, which takes into account the fact that the order of the handshake does not matter.
Q: Can I use a calculator to calculate the number of permutations and combinations?
A: Yes, you can use a calculator to calculate the number of permutations and combinations. However, it's always a good idea to understand the formulas and how to apply them manually.
Q: What are some real-world applications of permutations and combinations?
A: Permutations and combinations have many real-world applications, including:
- Scheduling events and activities
- Designing experiments and surveys
- Calculating probabilities and odds
- Optimizing business processes and operations
Q: Can I use permutations and combinations to solve other problems?
A: Yes, you can use permutations and combinations to solve many other problems. Some examples include:
- Calculating the number of ways to arrange objects in a specific order
- Finding the number of ways to select a subset of objects from a larger set
- Determining the probability of certain events occurring
Conclusion
In conclusion, permutations and combinations are fundamental concepts in mathematics that have many real-world applications. By understanding how to calculate permutations and combinations, you can solve a wide range of problems and make informed decisions in various fields. We hope this article has been helpful in answering your questions and providing a deeper understanding of these important mathematical concepts.
Final Answer
The final answer is: E. Combination; 14,365