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

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