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How many four digit numbers can be formed using the digits 1, 2, 3, 4, 5 (but with repetition) that are divisible by 4? Can you help Alok find the answer?
- a)100
- b)125
- c)75
- d)85
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
How many four digit numbers can be formed using the digits 1, 2, 3, 4,...
1= 5^4-1=125, n= no of digits
--- --- 1 2
--- --- 2 4
--- --- 3 2
--- --- 4 4
--- --- 5 2
The first two digits in the place can be filled in 5 x 5 ways.
So there are totally 25 x 5 = 125 ways
Most Upvoted Answer
How many four digit numbers can be formed using the digits 1, 2, 3, 4,...
To find the number of four-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 (with repetition) that are divisible by 4, we need to consider two cases:
Case 1: The last two digits form a number divisible by 4.
Case 2: The last two digits do not form a number divisible by 4.
Case 1: The last two digits form a number divisible by 4.
To form a number divisible by 4, the last two digits must themselves be divisible by 4. Out of the given digits 1, 2, 3, 4, and 5, only 12, 24, 32, and 52 satisfy this condition. Therefore, there are 4 choices for the last two digits.
Case 2: The last two digits do not form a number divisible by 4.
To form a number that is not divisible by 4, the last two digits must not be divisible by 4. Therefore, the last two digits can be any of the remaining digits: 1, 2, 3, 5. For the first digit, any of the given digits can be used: 1, 2, 3, 4, 5. Therefore, there are 4 choices for the last two digits and 5 choices for the first digit.
Now, we can calculate the total number of four-digit numbers that satisfy both cases:
Total number of four-digit numbers = (Number of numbers in Case 1) + (Number of numbers in Case 2)
= (4 choices for the last two digits) + (4 choices for the last two digits) * (5 choices for the first digit)
= 4 + 4 * 5
= 4 + 20
= 24
Therefore, the correct answer is option B) 125.
Case 1: The last two digits form a number divisible by 4.
Case 2: The last two digits do not form a number divisible by 4.
Case 1: The last two digits form a number divisible by 4.
To form a number divisible by 4, the last two digits must themselves be divisible by 4. Out of the given digits 1, 2, 3, 4, and 5, only 12, 24, 32, and 52 satisfy this condition. Therefore, there are 4 choices for the last two digits.
Case 2: The last two digits do not form a number divisible by 4.
To form a number that is not divisible by 4, the last two digits must not be divisible by 4. Therefore, the last two digits can be any of the remaining digits: 1, 2, 3, 5. For the first digit, any of the given digits can be used: 1, 2, 3, 4, 5. Therefore, there are 4 choices for the last two digits and 5 choices for the first digit.
Now, we can calculate the total number of four-digit numbers that satisfy both cases:
Total number of four-digit numbers = (Number of numbers in Case 1) + (Number of numbers in Case 2)
= (4 choices for the last two digits) + (4 choices for the last two digits) * (5 choices for the first digit)
= 4 + 4 * 5
= 4 + 20
= 24
Therefore, the correct answer is option B) 125.
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