Question

The number of natural numbers less than 1000 in which no digit is repeated is

Answer 1

Hint: We first break the total number of events in three parts. We find one digit, two digit or three-digit numbers with no repetition. We add them to find the final answer.

Complete step-by-step answer:
All the numbers less than 1000 are one digit, two digit or three-digit numbers.
We have to find these numbers whose digits aren’t repeated.
We first take one-digit natural numbers and there are 9 options 1 to 9.
Now we take two-digit numbers whose digits are not equal.
Two-digit number has two slots to fill-up. For a ten's place we can have numbers from 1 to 9. For the unit's place we can’t use the digit we already used in ten’s place but can use 0 instead.
Number of choices is $ 9\times 9=81 $ .
Now we take three-digit numbers whose digits are not equal.
Three-digit number has three slots to fill-up. For hundreds of places we can have numbers from 1 to 9. For ten’s place we can’t use the digit we already used in hundreds places but can use 0 instead. For unit’s place we have 8 choices.
Number of choices is $ 9\times 9\times 8=648 $ .
Therefore, the total number of choices are $ 648+81+9=738 $ .
So, the correct answer is “738”.

Note: The concept of no digits being repeated is equal to no two digits are equal. We can’t use any two same digits as all have to be unique.