How many four digit numbers have no repeat digits, do not co

Hmm. OK, lets start with the opposite of what is not allowed. We know that it's a 4 digit no and the sum of the digits has to be 28. Also, 4*7 = 28.
Let the no be 7777. However, no repetition is allowed.We try to re-arrange the given digits to form other acceptable 4 digit numbers.

Let's rearrange by reducing the value of the hundreds place by 1,and add it to the tens place : 7687. Again, take 2 units from the hundreds place and add it to the units place : 7489. All the conditions are met. Thus, as there are 4 distinct digits, the total no of arrangements possible: 4! = 24.

Again, lets try to rearrange the given number to see if we can end up with another such unique pair of combination : Reduce the value of the thousands place by 1 and add it to the hundreds place : 6589. Again, unique combination : 4! = 24.

Any other combination of value(s) will result in one of these 48 numbers.

Noteworthy point: The lowest of these numbers can only be of the form : 4XYZ.