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:

The value of \(P(n, r)\) increases as \(r\) increases up to a point, but the overall product depends on all factors. Notice that \(P(10, 4)\) is much larger than \(P(10, 2)\), which drives the result in this case.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves creating serial numbers with a specific format (letters and digits). Since the order matters and repetition is not allowed, this is a permutation problem. The total number of possibilities is the product of the number of ways to form the letter part and the number of ways to form the digit part.
Step 2: Key Formula or Approach:
The number of permutations of \(r\) objects taken from a set of \(n\) distinct objects is \(P(n, r) = \frac{n!}{(n-r)!}\).
Total possibilities = (Ways to arrange letters) \( \times \) (Ways to arrange digits).
Step 3: Detailed Explanation:
For Column A:
The format is 2 letters followed by 4 digits. - Letter Part: Choose and arrange 2 letters from 5 without repetition. \[ \text{Ways for letters} = P(5, 2) = 5 \times 4 = 20 \] - Digit Part: Choose and arrange 4 digits from 10 (0 to 9) without repetition. \[ \text{Ways for digits} = P(10, 4) = 10 \times 9 \times 8 \times 7 = 5040 \] - Total Serial Numbers: \[ \text{Total} = 20 \times 5040 = 100,800 \] So, Quantity A is 100,800.
For Column B:
The format is 4 letters followed by 2 digits. - Letter Part: Choose and arrange 4 letters from 5 without repetition. \[ \text{Ways for letters} = P(5, 4) = 5 \times 4 \times 3 \times 2 = 120 \] - Digit Part: Choose and arrange 2 digits from 10 (0 to 9) without repetition. \[ \text{Ways for digits} = P(10, 2) = 10 \times 9 = 90 \] - Total Serial Numbers: \[ \text{Total} = 120 \times 90 = 10,800 \] So, Quantity B is 10,800.
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 100,800
Quantity B = 10,800
Quantity A is greater than Quantity B.