Question

At a dog show, there are 20 judges and 10 dogs in the final round. Quantity \(\mathbf{A}\) The number of distinct pairs of judges Quantity \(\mathbf{B}\) The number of possible rankings of dogs from first to third place a. Quantity A is greater. b. Quantity \(\mathrm{B}\) is greater. c. The two quantities are equal d. The relationship cannot be determined from the information given.

Short answer

b. Quantity B is greater.

Step-by-step solution

Calculate Quantity A

Quantity A involves finding out the number of distinct pairs of 20 judges. As the order of the judges in a pair doesn't matter, we use combinations which is given by the formula \(\frac{n!}{k!(n-k)!}\). Here, \(n=20\) (total number of judges) and \(k=2\) (number of judges in each pair). So, Quantity A yields \(\frac{20!}{2!(20-2)!} = 190\).

Calculate Quantity B

Quantity B involves finding out the number of possible rankings of 3 dogs out of 10. In this scenario, order matters, so we use permutations. The formula for permutations is given by \(\frac{n!}{(n-k)!}\). Here, \(n=10\) (total number of dogs) and \(k=3\) (number of dogs being ranked). So, Quantity B yields \(\frac{10!}{(10-3)!} = 720\).

Compare Quantities

By comparing the two quantities: Quantity A is 190 and Quantity B is 720. Therefore, Quantity B is greater than Quantity A.

Key concepts

Combinatorics

Combinatorics is a fundamental area of mathematics dealing with counting, arranging, and combining objects. It is essential for solving problems where you need to quantify the arrangements or groupings of a certain set of items. In the context of the given exercise, combinatorics helps us determine how many different ways we can form pairs of judges from a larger group.

The specific tool we use here is called combinations, which is useful when the order of the items does not matter. For example, if you're selecting a pair of judges from a pool of 20, it doesn't matter if you pick judge A and judge B, or judge B and judge A. This symmetry in choice indicates the use of combinations rather than permutations.

• Combination formula: \( C(n, k) = \frac{n!}{k!(n-k)!} \)
• \( n \) is the total number of items to choose from (in our case, judges).
• \( k \) is the number of items to choose (here, pairs).

Applying this formula helps calculate exact counts where order doesn't affect the result. It's a critical skill for solving quantitative problems effectively on exams like the GRE.

Permutations

Permutations, unlike combinations, deal with arrangements where order matters. This is crucial when you want to know how many different ways you can arrange a subset of items, as is the case when ranking dogs by their performance.

In the exercise, we're tasked with ranking 3 dogs from a group of 10, where the sequence in which the dogs are ranked matters (first, second, and third place are distinct from each other). That's why permutations are the appropriate concept here.

• Permutation formula: \( P(n, k) = \frac{n!}{(n-k)!} \)
• \( n \) is the total number of dogs.
• \( k \) is the number of dogs being ranked (3 in this example).

Using this formula allows us to find the exact number of rankings possible, emphasizing how various orders or sequences can dramatically increase the count compared to combinations.

Quantitative Comparison

The GRE often includes questions that test your ability to compare different quantities. This skill involves mathematical reasoning to understand and compare numerical results obtained from different calculations.

In the provided problem, you have two different quantities: the number of combinations of judge pairs and the number of permutations of dog rankings. The exercise requires determining which of these outputs a larger value.

• Quantity A calculated using combinations gives us 190 pairs.
• Quantity B calculated using permutations gives us 720 rankings.

By comparing these values directly (190 vs 720), we conclude that more arrangements are possible in ranking the dogs, i.e., Quantity B is greater. Recognizing the appropriate mathematical approaches based on whether order matters (permutations) or not (combinations) is crucial for success in such quantitative comparison problems.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and applying mathematical principles to solve problems. It’s the backbone of the GRE's quantitative section, requiring students to not only perform calculations but also reason through the implications of these calculations.

To effectively tackle problems like the one given, you need to reason through the situation, identify the correct mathematical tools, and apply them correctly. This involves several steps:

• Understanding the problem context (e.g., forming pairs and ranking).
• Identifying whether combinations or permutations are needed.
• Accurately performing the calculations using specific formulas.
• Comparing and interpreting the results.

This type of reasoning helps in discerning between when two items are fundamentally different because of ordering or are the same when order is irrelevant, guiding how you select your combinatorial tools. A careful application of mathematical reasoning ensures clarity and precision in reaching the correct solution.