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:

When forming numbers, always be careful with the digit '0'. It cannot be placed in the leading position (e.g., the hundreds place of a 3-digit number). For questions asking for numbers "less than" a certain value (e.g., 1000), remember to sum the counts for all possible number lengths (1-digit, 2-digit, etc.).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem requires us to form numbers of different lengths (1-digit, 2-digit, 3-digit) using a given set of digits and certain rules, such as allowing repetition and handling the digit 0 correctly.
Step 2: Key Formula or Approach:
We use the fundamental principle of counting. We'll calculate the number of possibilities for each type of number (1-digit, 2-digit, 3-digit) and then sum them up for Column A.
Step 3: Detailed Explanation:
For Column A:
We need to form numbers less than 1000 using digits \{0, 2, 3, 4\} with repetition. This includes 1-digit, 2-digit, and 3-digit numbers.
1-digit numbers: The digits are \{2, 3, 4\}. The question states "0 can not be considered a one digit number". So there are 3 possible 1-digit numbers.
2-digit numbers: The tens place cannot be 0. So, it can be filled by \{2, 3, 4\} (3 ways). The units place can be any of the 4 digits \{0, 2, 3, 4\} (4 ways). Total 2-digit numbers = \(3 \times 4 = 12\).
3-digit numbers: The hundreds place cannot be 0. So, it can be filled by \{2, 3, 4\} (3 ways). The tens and units places can be any of the 4 digits. Total 3-digit numbers = \(3 \times 4 \times 4 = 48\).
Total numbers less than 1000 = (1-digit) + (2-digit) + (3-digit) = \(3 + 12 + 48 = 63\).
So, Quantity A is 63.
For Column B:
We need to form three-digit numbers using the digits \{1, 2, 3, 4\} with repetition allowed.
Hundreds place: Can be filled in 4 ways (1, 2, 3, or 4).
Tens place: Can be filled in 4 ways.
Units place: Can be filled in 4 ways.
Total three-digit numbers = \(4 \times 4 \times 4 = 4^3 = 64\).
So, Quantity B is 64.
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 63
Quantity B = 64
Therefore, the quantity on the right is greater.