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When dealing with permutations with restrictions (like "even number" and "no repetition"), it's often easiest to break the problem into cases based on the restricted position. Calculate the possibilities for each case and then add them up.
Updated On: Oct 3, 2025
- The quantity on the left is greater
- The quantity on the right is greater
- Both are equal
- The relationship cannot be determined without further information
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
We are forming two-digit even numbers from a given set of digits. The key difference between the two columns is whether repetition of digits is allowed or not. This will affect the number of choices available for the tens place.
Step 2: Key Formula or Approach:
We use the fundamental principle of counting. We determine the number of choices for the units place first (due to the "even" constraint), and then for the tens place.
Step 3: Detailed Explanation:
The set of available digits is \{2, 3, 4, 5\}. The even digits in this set are \{2, 4\}.
For Column A: Repetition is allowed
We need to form a two-digit even number. - Units place: Must be an even digit. We have 2 choices (2 or 4). - Tens place: Can be any of the 4 digits, since repetition is allowed. We have 4 choices (2, 3, 4, or 5). Total number of even numbers = (Choices for tens) \( \times \) (Choices for units)
\[ \text{Total ways} = 4 \times 2 = 8 \] The numbers are: 22, 32, 42, 52, 24, 34, 44, 54.
For Column B: Repetition is not allowed
We need to form a two-digit even number without repeating digits. It's best to consider cases based on the units digit.
Case 1: Units digit is 2.
- Units place has 1 choice (the digit 2). - Tens place cannot be 2. So, we have 3 choices left for the tens place (3, 4, or 5). - Number of ways for this case = \( 3 \times 1 = 3 \). (Numbers: 32, 42, 52)
Case 2: Units digit is 4.
- Units place has 1 choice (the digit 4). - Tens place cannot be 4. So, we have 3 choices left for the tens place (2, 3, or 5). - Number of ways for this case = \( 3 \times 1 = 3 \). (Numbers: 24, 34, 54)
Total number of even numbers = (Ways from Case 1) + (Ways from Case 2)
\[ \text{Total ways} = 3 + 3 = 6 \] Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 8
Quantity B = 6
Therefore, Quantity A is greater than Quantity B.
We are forming two-digit even numbers from a given set of digits. The key difference between the two columns is whether repetition of digits is allowed or not. This will affect the number of choices available for the tens place.
Step 2: Key Formula or Approach:
We use the fundamental principle of counting. We determine the number of choices for the units place first (due to the "even" constraint), and then for the tens place.
Step 3: Detailed Explanation:
The set of available digits is \{2, 3, 4, 5\}. The even digits in this set are \{2, 4\}.
For Column A: Repetition is allowed
We need to form a two-digit even number. - Units place: Must be an even digit. We have 2 choices (2 or 4). - Tens place: Can be any of the 4 digits, since repetition is allowed. We have 4 choices (2, 3, 4, or 5). Total number of even numbers = (Choices for tens) \( \times \) (Choices for units)
\[ \text{Total ways} = 4 \times 2 = 8 \] The numbers are: 22, 32, 42, 52, 24, 34, 44, 54.
For Column B: Repetition is not allowed
We need to form a two-digit even number without repeating digits. It's best to consider cases based on the units digit.
Case 1: Units digit is 2.
- Units place has 1 choice (the digit 2). - Tens place cannot be 2. So, we have 3 choices left for the tens place (3, 4, or 5). - Number of ways for this case = \( 3 \times 1 = 3 \). (Numbers: 32, 42, 52)
Case 2: Units digit is 4.
- Units place has 1 choice (the digit 4). - Tens place cannot be 4. So, we have 3 choices left for the tens place (2, 3, or 5). - Number of ways for this case = \( 3 \times 1 = 3 \). (Numbers: 24, 34, 54)
Total number of even numbers = (Ways from Case 1) + (Ways from Case 2)
\[ \text{Total ways} = 3 + 3 = 6 \] Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 8
Quantity B = 6
Therefore, Quantity A is greater than Quantity B.
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