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Let n be the number of different five-digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?
- a)144
- b)168
- c)192
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let n be the number of different five-digit numbers, divisible by 4 wi...
To be divisible by 4 , last 2 digits of the 5 digit no. should be divisible by 4.
So possibilities are 12,16,32,64,24,36,52,56 which are 8 in number.
Remaining 3 digits out of 4 can be selected in ways and further can be arranged in 3! ways .
So in total = 8*4*6 = 192
So possibilities are 12,16,32,64,24,36,52,56 which are 8 in number.
Remaining 3 digits out of 4 can be selected in ways and further can be arranged in 3! ways .
So in total = 8*4*6 = 192
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Community Answer
Let n be the number of different five-digit numbers, divisible by 4 wi...
Understanding the Problem
To find the number of different five-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6, and are divisible by 4, we need to focus on two main points:
- The last two digits of the number must form a number that is divisible by 4.
- All digits must be unique, drawn from the set {1, 2, 3, 4, 5, 6}.
Divisibility Rule for 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. We will first list all the valid two-digit combinations from the given digits:
- Valid pairs: 12, 16, 24, 32, 36, 52, 56, 64
These pairs (12, 16, 24, 32, 36, 52, 56, 64) must be examined to ensure they can be formed from our digit set without repetition.
Counting Valid Combinations
1. Count Valid Pairs:
- From the above pairs, we have 8 valid combinations: 12, 16, 24, 32, 36, 52, 56, 64.
2. Calculate Remaining Digits:
- For each valid pair, there are 4 remaining digits to choose for the first three positions.
3. Arrangements for First Three Digits:
- Each of the 4 remaining digits can be arranged in 3! (6) ways.
Calculating Total Combinations
- For each of the 8 valid pairs, we can form:
- 4 remaining digits × 6 arrangements = 24 combinations.
- Thus, the total number of five-digit numbers is:
- 8 pairs × 24 combinations = 192.
Conclusion
The total number of different five-digit numbers that can be formed is:
- n = 192
Thus, the answer is option C: 192.
To find the number of different five-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6, and are divisible by 4, we need to focus on two main points:
- The last two digits of the number must form a number that is divisible by 4.
- All digits must be unique, drawn from the set {1, 2, 3, 4, 5, 6}.
Divisibility Rule for 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. We will first list all the valid two-digit combinations from the given digits:
- Valid pairs: 12, 16, 24, 32, 36, 52, 56, 64
These pairs (12, 16, 24, 32, 36, 52, 56, 64) must be examined to ensure they can be formed from our digit set without repetition.
Counting Valid Combinations
1. Count Valid Pairs:
- From the above pairs, we have 8 valid combinations: 12, 16, 24, 32, 36, 52, 56, 64.
2. Calculate Remaining Digits:
- For each valid pair, there are 4 remaining digits to choose for the first three positions.
3. Arrangements for First Three Digits:
- Each of the 4 remaining digits can be arranged in 3! (6) ways.
Calculating Total Combinations
- For each of the 8 valid pairs, we can form:
- 4 remaining digits × 6 arrangements = 24 combinations.
- Thus, the total number of five-digit numbers is:
- 8 pairs × 24 combinations = 192.
Conclusion
The total number of different five-digit numbers that can be formed is:
- n = 192
Thus, the answer is option C: 192.
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Let n be the number of different five-digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?a)144b)168c)192d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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