How many eight-digit numbers can be formed under each condition?
a) The leading digit cannot be zero, the fifth digit cannot be 6 or 8, and the number must be less then 75,000,000.
b) The leading digit cannot be zero, the number must be divisible by 5, the fourth digit cannot be 2, and no repetition of digits is allowed.
The Attempt at a Solution
So because you have 8 spaces, and any number between 0-9 can fit, you have a total of 10 choices you can put in a single spot.
_ _ __ __ _ __ __ __
7 5 10 10 8 10 10 10
Multiplied together = 28,000,000 different eight-digit combinations.
The number has to be lower then 75,000,000 so highest number you can have there is a 7, meaning 8, but no 0 either, so 7. The next number can only be up to a 4, so you have 5 choices between 0-4. The next two spots are 10's because they have no parameters. The fifth digit is a 8 because you can't have the numbers 6 or 8 there. And the rest are 10's.
I tried doing this based of the number: 74,999,999 so it would be less then 75,000,000 and fill the rest based off of that.
_ _ _ _ _ _ _ _
1 9 8 6 6 5 4 3
Multiplied together = 155,520 different choices
For the first digit, it can't be a 0, so it takes the choices for that spot to 1-9, and the number has to be divisible by 5, so there is basically only one choice, of 5, then the next number is 9 because no repetition is allowed and same for the 8. The fourth digit can't be a 2, so instead of 7, it lowers to 6. The next digit is 6 too because you still have that many choices. And then it decreases one each time because of the no repetition.
I think I did them right, but I am not sure, that is why I am looking for help. Thanks!