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:

Look for keywords that indicate order matters, such as "arrange", "one after the other", "rank", "signal", or assigning distinct roles. If order matters, it's a permutation. If the wording was just "select a group of 3", it would be a combination.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Both problems describe situations where we need to select a subset of items from a larger set and arrange them in a specific order. The phrases "one below the other" and "one after the other" indicate that the order of selection is important. Therefore, both are permutation problems.
Step 2: Key Formula or Approach:
The number of arrangements (permutations) of \(r\) objects taken from a set of \(n\) distinct objects is given by the formula: \[ P(n, r) = \frac{n!}{(n-r)!} = n \times (n-1) \times \dots \times (n-r+1) \] Step 3: Detailed Explanation:
For Column A:
We are selecting 3 flags from 7 distinct flags and arranging them in a vertical order. This is a permutation of 3 items from a set of 7. Here, \(n=7\) and \(r=3\). Number of different signals = \(P(7, 3)\). \[ P(7, 3) = 7 \times 6 \times 5 = 210 \] Alternatively: - The top position can be filled by any of the 7 flags. - The middle position can be filled by any of the remaining 6 flags. - The bottom position can be filled by any of the remaining 5 flags. Total ways = \(7 \times 6 \times 5 = 210\).
For Column B:
We are selecting 3 children from 7 distinct children and arranging them in a line. This is also a permutation of 3 items from a set of 7. Here, \(n=7\) and \(r=3\). Number of ways to arrange the children = \(P(7, 3)\). \[ P(7, 3) = 7 \times 6 \times 5 = 210 \] Alternatively: - The first position can be filled by any of the 7 children. - The second position can be filled by any of the remaining 6 children. - The third position can be filled by any of the remaining 5 children. Total ways = \(7 \times 6 \times 5 = 210\).
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 210
Quantity B = 210
The two quantities are equal.