Question

How many numbers lying between 10 and 1000 can be formed from the digits 1,2,3,4,5,6,7,8,9 (repetition is allowed)
A. 1024
B. 810
C. 2346
D. None of these

Answer 1

Hint: In this question you are given a range of numbers 10-1000 and you need to find the total number of numbers which can be formed by using digits 1,2,3,4,5,6,7,8,9. So, divide the solution in two parts, first find the total number of two digit numbers which are greater than 10 then find the total number of three digit numbers. Then add the two to get the required answer to the problem.

Complete step-by-step answer:
Total number of two digit numbers using 1,2,3,4,5,6,7,8,9 = $9\times 9$ (since repetition is allowed)
So total two-digit numbers = 81
We do not consider the numbers 10,20 ….90 as we are not supposed to use 0 here,hence its 9x9 and not 10x9.
Total number of three-digit numbers using 1,2,3,4,5,6,7,8,9= $9\times 9\times 9$
So, total three-digit numbers = 729
Total number of numbers lying between 10 and 1000 which can be formed from the digits 1,2,3,4,5,6,7,8,9 = Total two-digit numbers + Total three-digit numbers
= 81+729
=810
So there are 810 numbers between 10-1000 which can be formed by using 1,2,3,4,5,6,7,8,9.
Thus, option B. is correct.

Note: In these types of questions we use the fundamental principle of counting which states that if there are p ways to do one thing, and q ways to do another, then there are p $\times$ q ways to do both things. This principle can also be used when there are more than two choices involved in the experiment.