Question

• A. The quantity on the left is greater
• B. The quantity on the right is greater
• C. Both are equal
• D. The relationship cannot be determined without further information

Hint:

In quantitative comparison, if one of the quantities cannot be calculated due to missing information, the answer is almost always (D). Be alert for vague terms like "students of a school" without specifying a number.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Column A presents a clear permutation problem. We are distributing distinct prizes, so the order matters, and each boy can get at most one. Column B is ambiguous. It lacks crucial information needed for a calculation, such as the number of students, whether prizes are distinct, and if a student can receive more than one prize.
Step 2: Key Formula or Approach:
For Column A, the number of ways to arrange \(r\) items from a set of \(n\) is given by the permutation formula: \[ P(n, r) = \frac{n!}{(n-r)!} \] Step 3: Detailed Explanation:
For Column A:
We are distributing 3 prizes among 5 boys, with the condition that no boy gets more than one prize. We assume the prizes are distinct (e.g., 1st, 2nd, 3rd).
The first prize can be given to any of the 5 boys.
The second prize can be given to any of the remaining 4 boys.
The third prize can be given to any of the remaining 3 boys.
Total number of ways = \( 5 \times 4 \times 3 = 60 \).
This is equivalent to the permutation \(P(5, 3)\): \[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 5 \times 4 \times 3 = 60 \] For Column B:
We are asked for the number of ways to distribute 5 prizes among "the students of a school".
This statement is not specific enough to perform a calculation because:
1. Number of students is unknown: Let the number of students be \(n\). We don't know the value of \(n\). Is \(n<5\), \(n = 5\), or \(n>5\)?
2. Nature of prizes is unknown: Are the 5 prizes distinct or identical?
3. Distribution rules are unknown: Can one student receive more than one prize?
Because we cannot calculate a specific value for Quantity B, we cannot compare it to Quantity A. For example, if there are only 2 students, the number of ways is \(2^5=32\). If there are 10 students and no student can get more than one prize, the ways are \(P(10,5) = 30240\). The result changes based on the assumptions.
Step 4: Final Answer:
Quantity A = 60.
Quantity B cannot be determined from the given information.
Therefore, the relationship between Quantity A and Quantity B cannot be determined.