Q. 254.7( 3 Votes )

How many three-digit numbers are there, with no digit repeated?

Answer :

Given, digits which can be used to make numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9


The number of these digits are 10


To find: Total number of three-digit numbers with no digit repeated


Formula used:


Number of arrangements of n things taken r at a time = P(n, r)



The total number of ways


= the number of arrangements of 10 things taken 3 at a time


= P(10, 3)





= 10 × 9 × 8


= 720


But in these 720 numbers we also included those numbers which are starting from 0 like 023 or 056 etc. Being starting from 0, these are actually two-digit numbers. So, we need to subtract these numbers.


To find these numbers, fix the position of 0 at hundred’s place.



Remaining numbers = 9 (1, 2, 3, 4, 5, 6, 7, 8 or 9)


Arrange these 9 numbers in remaining 2 places.


Formula used:


Number of arrangements of n things taken r at a time = P(n, r)



The total numbers which are starting from 0 are =


= the number of arrangements of 9 things taken 2 at a time


= P(9, 2)





= 9 × 8


= 72


Hence, total number of three-digit numbers with no digit repeated are, 720 – 72 = 648


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