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How many even numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6 (repetitions of digits are allowed)?
- a)648
- b)180
- c)1296
- d)600
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
How many even numbers of four digits can be formed with the digits 1, ...
The number of numbers formed would be given by 5 x 4 x 3 (given that the first digit can be filled in 5 ways, the second in 4 ways and the third in 3 ways - MNP rule).
Most Upvoted Answer
How many even numbers of four digits can be formed with the digits 1, ...
To form a four-digit number, we need to consider the following cases:
Case 1: The thousands digit is even.
In this case, we have three choices for the thousands digit: 2, 4, or 6. After selecting the thousands digit, we have six choices for each of the remaining three digits. Thus, the total number of possibilities for this case is 3 * 6 * 6 * 6 = 648.
Case 2: The thousands digit is odd.
In this case, we have two choices for the thousands digit: 1 or 3. After selecting the thousands digit, we have six choices for each of the remaining three digits. Thus, the total number of possibilities for this case is 2 * 6 * 6 * 6 = 432.
Therefore, the total number of even four-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 is 648.
Explanation of each case:
1. The Thousands Digit is Even:
- In this case, we have three choices for the thousands digit: 2, 4, or 6.
- After selecting the thousands digit, we have six choices for each of the remaining three digits.
- This is because we can choose any of the digits 1, 2, 3, 4, 5, or 6 for each of the units, tens, and hundreds places.
- Thus, the total number of possibilities for this case is 3 * 6 * 6 * 6 = 648.
2. The Thousands Digit is Odd:
- In this case, we have two choices for the thousands digit: 1 or 3.
- After selecting the thousands digit, we have six choices for each of the remaining three digits.
- This is because we can choose any of the digits 1, 2, 3, 4, 5, or 6 for each of the units, tens, and hundreds places.
- Thus, the total number of possibilities for this case is 2 * 6 * 6 * 6 = 432.
Total Number of Possibilities:
- To find the total number of even four-digit numbers, we add the possibilities from both cases.
- Therefore, the total number of even four-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 is 648 (648 + 432 = 1080).
Case 1: The thousands digit is even.
In this case, we have three choices for the thousands digit: 2, 4, or 6. After selecting the thousands digit, we have six choices for each of the remaining three digits. Thus, the total number of possibilities for this case is 3 * 6 * 6 * 6 = 648.
Case 2: The thousands digit is odd.
In this case, we have two choices for the thousands digit: 1 or 3. After selecting the thousands digit, we have six choices for each of the remaining three digits. Thus, the total number of possibilities for this case is 2 * 6 * 6 * 6 = 432.
Therefore, the total number of even four-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 is 648.
Explanation of each case:
1. The Thousands Digit is Even:
- In this case, we have three choices for the thousands digit: 2, 4, or 6.
- After selecting the thousands digit, we have six choices for each of the remaining three digits.
- This is because we can choose any of the digits 1, 2, 3, 4, 5, or 6 for each of the units, tens, and hundreds places.
- Thus, the total number of possibilities for this case is 3 * 6 * 6 * 6 = 648.
2. The Thousands Digit is Odd:
- In this case, we have two choices for the thousands digit: 1 or 3.
- After selecting the thousands digit, we have six choices for each of the remaining three digits.
- This is because we can choose any of the digits 1, 2, 3, 4, 5, or 6 for each of the units, tens, and hundreds places.
- Thus, the total number of possibilities for this case is 2 * 6 * 6 * 6 = 432.
Total Number of Possibilities:
- To find the total number of even four-digit numbers, we add the possibilities from both cases.
- Therefore, the total number of even four-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 is 648 (648 + 432 = 1080).
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Community Answer
How many even numbers of four digits can be formed with the digits 1, ...
The answer should be 60, i.e., 5x4x3
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How many even numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6 (repetitions of digits are allowed)?a)648b)180c)1296d)600Correct answer is option 'A'. Can you explain this answer?
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