Four digit numbers with no repeats

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Question from Daniel, a parent:

I want to know how many different numbers there would be between the numbers
0000-9999 that do not have any numbers that duplicate or repeat in the number? For example
the first number would have to be 0123 and the last would be 9876 (I think) Thanks!

Daniel,

How many one digit numbers can you form with no repeats? That's clear, there are 10. They are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

How many two digit numbers can you form with no repeats?

Each of the 10 one digit numbers in the list above can be extended to a two digit number in 9 ways. Thus there are 10 × 9 two digit numbers with no repeats. They are

01
02
03
.
.
.
09
10
12
.
.
.
98

How many three digit numbers can you form with no repeats?

Each of the 90 two digit numbers in the list above can be extended to a three digit number in 8 ways. Thus there are 10 × 9 × 8 three digit numbers with no repeats.

How many four digit numbers can you form with no repeats?

Penny

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