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 prize distribution problems, pay close attention to whether the prizes are distinct or identical. "1st, 2nd, 3rd" implies distinct prizes (use permutations). "Consolation prizes" or "prizes of the same value" imply identical prizes (use combinations).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Column A is a two-step counting problem. The phrase "first three" implies that the prizes are distinct (1st, 2nd, 3rd), so order matters (a permutation). The "two consolation prizes" are typically identical, so order does not matter (a combination). We need to calculate this value and compare it to the product in Column B.
Step 2: Key Formula or Approach:
Number of ways to arrange \(r\) items from \(n\) is \(P(n, r) = \frac{n!}{(n-r)!}\).
Number of ways to choose \(r\) items from \(n\) is \(C(n, r) = \frac{n!}{r!(n-r)!}\).
We use the multiplication principle to combine the two steps.
Step 3: Detailed Explanation:
For Column A:
Step i: Give the first three prizes to 3 students from 20. Since the prizes are for 1st, 2nd, and 3rd place, they are distinct, and the order matters. \[ \text{Ways to give first three prizes} = P(20, 3) = 20 \times 19 \times 18 \] Step ii: After giving the first three prizes, \(20 - 3 = 17\) students remain. We need to give two identical consolation prizes to 2 of these 17 students. Since the prizes are identical, the order of selection does not matter. \[ \text{Ways to give consolation prizes} = C(17, 2) = \frac{17 \times 16}{2 \times 1} = 17 \times 8 \] Step iii: The total number of ways is the product of the ways in the two steps. \[ \text{Total Ways} = P(20, 3) \times C(17, 2) = (20 \times 19 \times 18) \times (17 \times 8) \] Quantity A = \( 20 \times 19 \times 18 \times 17 \times 8 \).
For Column B:
We are given a product of numbers: \( 2 \times 2 \times 5 \times 19 \times 3 \times 3 \times 2 \times 17 \times 2 \times 2 \times 2 \times 2 \) Let's group these factors to make them comparable to Column A. \[ (2 \times 2 \times 5) = 20 \] \[ 19 \] \[ (3 \times 3 \times 2) = 18 \] \[ 17 \] \[ (2 \times 2 \times 2 \times 2) = 16 \] So, Quantity B = \( 20 \times 19 \times 18 \times 17 \times 16 \).
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = \( 20 \times 19 \times 18 \times 17 \times 8 \)
Quantity B = \( 20 \times 19 \times 18 \times 17 \times 16 \)
Since all other factors are the same, we just compare 8 and 16. As \(16>8\), Quantity B is greater than Quantity A.