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

Hint: 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.