How many 4 digit numbers can be formed using a digit only once per number?

Answer 1

There are ten digits 0-9 that you can choose from. 10P4 = is the number of ways to order these ten digits into four digit numbers.

However, 0 cannot begin the number or else it is a 3 digit number (0341 is a three digit number). So, we must subtract from 5040 the number of 3 digit numbers which do not contain 0. There are 1-9 digits to choose from. 9P3 = 504 is the number of ways to order these nine digits into three digit numbers.

5040 - 504 = 4536

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Another way to arrive at the same result:

You have 9 choices for the first number (1,2,3,...,9). Once you choose the first number you have 9 numbers to choose from for each of the remaining three numbers: (0,1,2,3,...9) but excluding the number you chose for the first number. So you have (9P3) ways to choose the final three digits.

Thus, you have a total of 9*(9P3) ways to form your four digit number:

9*(9P3)=9*9*8*7=4536