You didn't specify whether we may repeat digits or not.
I will assume that we cannot. If we can then post again
stating that digits may be repeated.
Assuming digits cannot be repeated:
a) the numbers must be less than 500?
Counting 1 and 2 digit numbers;
We consider 1 digit numbers as 2-digit numbers with
left-most digit 0.
Choose units digit 8 ways {0,1,2,3,4,5,6,7}
Choose tens digit 7 ways
That's 8*7 = 56 1 and 2 digit numbers, counting 0.
Counting three digit numbers:
Choose hundreds digit 4 ways {1,2,3,4}
Choose tens digit 7 ways.
Choose units digit 6 ways
That's 4*7*6 = 168
So that's 56+168 = 224 counting 0.
-------------------------
b) the numbers must be odd?
I suppose you mean it can have any number of digits up through 8 digits,
since you didn't state otherwise.
7 or 8 digit numbers;
(We consider 7 digit numbers as 8 digit numbers whose left-most
digit is 0.)
Choose units digit 4 ways {1,3,5,7}
Choose tens-digit 7 ways
Choose hundreds digit 6 ways
Choose thousands digit 5 ways
Choose ten thousands digit 4 ways
Choose hundred thousands digit 3 ways
Choose millions digit 2 ways
Choose ten millions digit 1 way
That's 4*7*6*5*4*3*2*1
------------
6 or 5 digit numbers
(We consider 5 digit numbers as 6 digit numbers whose left-most
digit is 0.)
Choose units digit 4 ways {1,3,5,7}
Choose tens-digit 7 ways
Choose hundreds digit 6 ways
Choose thousands digit 5 ways
Choose ten thousands digit 4 ways
Choose hundred thousands digit 3 ways
That's 4*7*6*5*4*3
----------------------
Similarly, the number of 3 or 4 digit numbers is 4*7*6*5
----------------------
Similarly, the number of 1 or 2 digit numbers is 4*7 = 28
----------------------
Total: 4*7*6*5*4*3*2*1 + 4*7*6*5*4*3 + 4*7*6*5 + 4*7 = 31108
-------------------
c) the numbers must be even?
It's the same as (b) since there are same number even digits
{0,2,4,6) and there are odd digits {1,3,5,7} for the units
digit. Also 31108.
Edwin
