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29 Mar 2020, 22:09
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In how many ways, you can form a 4 digit number using the digits 0 to 9 without repetition, Such that the unit dig is multiple of 3 , the 10th and 100th dig are factors of 3 and 1000th dig is divisible of 3?
31 Mar 2020, 00:20
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Hello Bijay,
I recommend that you practice questions where the question is framed better; this way, you will not have to bear the added burden of comprehending a question whose language is, at best, confusing.
Nevertheless, I seem to have got an overall idea of what the question is trying to ask. We are to form a 4-digit number with distinct digits, using the digits 0 to 9 such that the units digit and the thousands digit are multiples of 3 while the tens and the hundreds digit are factors of 3.
This means the units place can be filled by the digits 0 or 3 or 6 or 9 (remember that 0 is a multiple of every number); the thousands place can be filled by the digits 3 or 6 or 9.
The hundreds and the tens place can be filled by the digits 1 or 3 since these are the only 2 factors of 3.
Since the number 3 HAS to be placed in either the tens or hundreds place, we can conclude that we cannot place 3 in the thousands or the units place. Therefore, the thousands place can be filled by 6 or 9 while the units place can be filled by 0 or 6 or 9.
When the units place is filled by 0, the thousands place can be filled by 6 or 9. So, we can form 4 numbers viz., 6310, 6130, 9310 and 9130.
When the units place is filled by 6, the thousands place can be filled by 9 and hence the numbers can be 9136 and 9316.
When the units place is filled by 9, the thousands place can be filled by 6 and hence the numbers can be 6139 and 6319.
We can see that there are a total of 8 numbers possible. IMO, this represents the total number of ways in which 4 digit numbers can be formed satisfying the given constraints.
In a question like this with constraints, you will have to take care of the constraints first before you deal with the entities on which there are no constraints. In this case, it was important that we understood that the tens and the hundreds digit could only be filled by 1 and 3 which would automatically exclude these digits from appearing in the thousands and the units places.
Hope that helps!
I recommend that you practice questions where the question is framed better; this way, you will not have to bear the added burden of comprehending a question whose language is, at best, confusing.
Nevertheless, I seem to have got an overall idea of what the question is trying to ask. We are to form a 4-digit number with distinct digits, using the digits 0 to 9 such that the units digit and the thousands digit are multiples of 3 while the tens and the hundreds digit are factors of 3.
This means the units place can be filled by the digits 0 or 3 or 6 or 9 (remember that 0 is a multiple of every number); the thousands place can be filled by the digits 3 or 6 or 9.
The hundreds and the tens place can be filled by the digits 1 or 3 since these are the only 2 factors of 3.
Since the number 3 HAS to be placed in either the tens or hundreds place, we can conclude that we cannot place 3 in the thousands or the units place. Therefore, the thousands place can be filled by 6 or 9 while the units place can be filled by 0 or 6 or 9.
When the units place is filled by 0, the thousands place can be filled by 6 or 9. So, we can form 4 numbers viz., 6310, 6130, 9310 and 9130.
When the units place is filled by 6, the thousands place can be filled by 9 and hence the numbers can be 9136 and 9316.
When the units place is filled by 9, the thousands place can be filled by 6 and hence the numbers can be 6139 and 6319.
We can see that there are a total of 8 numbers possible. IMO, this represents the total number of ways in which 4 digit numbers can be formed satisfying the given constraints.
In a question like this with constraints, you will have to take care of the constraints first before you deal with the entities on which there are no constraints. In this case, it was important that we understood that the tens and the hundreds digit could only be filled by 1 and 3 which would automatically exclude these digits from appearing in the thousands and the units places.
Hope that helps!
