Question from Katy, a teacher:
Using only the digits 0,3,4,5,6,and 7, how many distinct four-digit numbers exist that are greater than 4002 and less than 6732?
Let's break it down into chunks. Clearly the valid numbers all start
with a 4, 5 or 6:
- If it starts with a 5, any following digits are allowed from the selection 0, 3, 4, 5, 6, 7 for the remaining three places (hundreds, tens, ones). There are 6 choices for each of the 3 places. That means 63 distinct valid numbers starting with a 5.
- If it starts with a 4, then we have the same 63 choices with only one exception: 4000 can be made with the right digits, but isn't greater than 4002. Therefore there are 63 - 1 distinct valid numbers starting with a 4.
- If it starts with a 6, it gets more complicated. It may start with a 60, 63, 64, 65, 66, or 67. Only the 67 needs special treatment, so let's put it aside for now. The others (there are 5 choices) can have any of the normal 6 digits in the tens and any of them in the ones column. That's 62 for each of the 5 two-digit prefixes.
Now let's get back to the numbers starting with 67. In this case, it is easy to just list them and count them: 6700, 6703 6704 6705 6706 6707 6730. That's it: 7 distinct valid numbers starting with 67 which are less than 6732.
Finally, add them up to get the total:
 + [63 - 1] + [5 x 62 + 7]
Hope this helps,
Stephen La Rocque.