Find number of four-digit numbers in which repetition is not allowed. Also find number of four-digit numbers in which at least one digit is repeated.

Explanation

Let us first create four places.

We start at thousands place. We have 9 options for this place. Since repetition of digits is not allowed, at hundred's place we cannot use digit which is used in thousand's place.
So, for this place, we have 8 options ad digits used earlier cannot be used.
Subsequently for unit place we have 7 options.
So, total number of numbers `=9xx9xx8xx7=4536`
In such problem, it is important from which place you start. If you start at unit place, you count 10 options for this place. Then for ten's and hundred's place you count 9 and 8 options respectively. Now for thousand's place number of options depends upon whether digit '0' is used or not. If '0' is used, you have 7 options and if '0' is not used you have 6 options for this place. Here it will be difficult to make cases, Hence, starting at thousands place is good idea.
Now number of four-digit numbers in number system is 9000. So, number of numbers in which at least one digit is repeated =total four-digit numbers- number of numbers in which all the digits are distinct
=9000-4536
=4464