Question

How many 4-digit numbers are there when a digit may be repeated any number of times?
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Answer 1

Hint:In this question, we have to form 4-digit numbers. In the thousand’s place, we have to take any digit from 1,2,3,4,5,6,7,8, and 9. So, we have 9 intakes in the thousand’s place. In hundred’s place, we can take any digit from 0,1,2,3,4,5,6,7,8, and 9. So, we have 10 intakes in hundred’s place. Similarly, for ten’s place and one’s place, we have 10 intakes. By multiplying all these results, we will be able to get the final answer.

Complete step-by-step answer:
We have to find the number of 4-digit numbers. For that, we have to first find the number of intakes for each place of 4 digit number.
In the thousand’s place, we have to choose any number from 1,2,3,4,5,6,7,8, and 9. We have a total of 9 intakes for the thousand’s place of 4- digit numbers.
Now, again for the hundred’s place, we have to find the number of intakes. Hundred’s place can have any digit from 0,1,2,3,4,5,6,7,8, and 9. So we have 10 intakes for the hundred’s place.
For the ten’s place, we also have to find the number of intakes. Ten’s place can have any digit from 0,1,2,3,4,5,6,7,8, and 9. So, we have a total of 10 intakes for ten’s place also.
Similarly, for the one’s place, we have 10 intakes.
Now, understand this with a diagram.

Multiplying the intakes of thousand’s place, hundred’s place, ten’s place, and one’s place, we can get the total number of 4- digit numbers.
Hence, the total number of 4-digit numbers \[=9\times 10\times 10\times 10=9\times {{10}^{3}}=9000\] .

Note:In this question, the common mistake that can be done is taking “0” in the thousand’s place. As numbers consist of any digit from 0 to 9. But we cannot take 0 in the thousand’s place. If we do so then our number will be a three-digit number. But we have to form a 4-digit number. So, we have to ignore “0” to be included in the thousand’s place.