Question

How many numbers lying between $99$ and $1000$ can be made from the digits $2,3,7,0,8,6$ when the digits occur only once in each number?
$A)100$
$B)90$
$C)120$
$D)80$

Answer 1

First, from the given that we have numbers between $99$ and $1000$ . These numbers need to be made from the given digits $2,3,7,0,8,6$ .
Condition is given as digits occurring only once in each number means digits without repetition in the process of finding the numbers between $99$ and $1000$ .
Between the numbers, $99,1000$ will not be included.

Complete step by step answer:
Since from the given that we have numbers between $99$ and $1000$ . There are only three digits numbers because $99,1000$ will not be included. Thus, we reframe the given question from the numbers $100$ to $999$ .
Now we have to find the number of ways made from the given digits $2,3,7,0,8,6$ for the numbers $100$ to $999$ .
Place Hundred:
For the place hundred, the starting digit will not be zero (because if it's zero then it will be placed ten) and thus we get the non-zero numbers $2,3,7,8,6$ are the number of ways.
Hence, we get a total $5$ ways can be made in a hundred places.
Place Ten:
For place ten, we have a total $6$ possible ways, but the condition is given as digits occurring only once in each number means digits without repetition.
So, in the place of a hundred occurrences (that one number) will not be repeated.
Thus, we have $4 + 1$ (in hundred places not occurred numbers plus the number zero) ways
Hence, we have a total $5$ ways that can be made in tenth place.
Unit place:
Again, for the unit place, the possible ways are $6$ .
But in hundred places occurred digit and also in the tenth place occurred digits are not be repeated and thus we have $6 - 2 = 4$ ways for the unit place.
Therefore, in total we have $5 \times 5 \times 4 = 100$ ways.

So, the correct answer is “Option A”.

Note: Suppose the given question is about lying from $99$ and $1000$ be made from the digits $2,3,7,0,8,6$ then we also need to find the place values for two digits occurring $99$ and four digits occurring $1000$ .
Repetition means once previously used numbers can be repeated and digits occur only once in each number means no repetition and the number used will not be repeated.