Using the digits 0-9 find out how many 4 digit numbers can be configured, based on the stated conditions

Question

 

 

 

Using the digits 0-9 find out how many 4 digit numbers can be configured, based on the stated conditions:

 

A. The number must be at least 5000 and be divisible by 10. (Repeated digits are okay.)

 

B. The number must be less than 3000 and must be even. No digits may be repeated in

the last 3 digits. (That is, 2234 would be okay, but 2334 would not be okay.)

Answer 1

Using the digits 0-9 find out how many 4 digit numbers can be configured, based on the stated conditions:

A. The number must be at least 5000 and be divisible by 10. (Repeated digits are okay.)

"at least 5000" means the number goes from 5000 to 9999, but "divisible by 10" means the form of the number is xxx0, so the last digit is always 0. This results in a 4-digit counter like this:

    A   B    C   D

  ---  ---  ---  ---      [choose one number from each column]

    5   0    0    0          [like an odometer counts from 0 to 9]

    6   1    1

    7   2    2

    8   3    3

    9   4    4

         5    5

         6    6

         7    7

         8    8

         9    9

This is every tenth number from 5000 to 9990:  5*10*10*1 = 500 of them.

B. The number must be less than 3000 and must be even. No digits may be repeated in

the last 3 digits. (That is, 2234 would be okay, but 2334 would not be okay.)

 

Now, the first digit (A) is restricted to 0-2, the last digit (D) may be any of (0,2,4,6,8), but once digit D is selected, it may not be used as either digit B or digit C.  That lowers the number of choices.

 

So, there are 3 possibilities for A.  Then there are 5 possibilities for D.  Then there are 9 (no longer 10) possibilities for B and 8 (no longer 10 or 9 since "No digit may be repeated in the last 3 digits") possibilities for C.

 

That makes 3*5*9*8 = 1080 numbers.

     [Note: 4-digit numbers with leading zeros, like 0120, are o.k.]