How many numbers are there which have five digits, each having a number in $\{1,2,3,4,5,6,7,8,9\}$ and either having all odd or all even?
Solution:
So there are 5 digits in that set that are odd, and 4 that are even
Case 1: all odd
we would have $5*5*5*5*5=5^5$
Case 2: all even
we would have $4*4*4*4*4 = 4^5$
Therefore combining both cases we have $5^5+4^5$
I am wondering if this logic is correct as it is my first few problems solved like this.
You must be assuming that the numbers may be used repeatedly, else you would not have $5$ distinct digits that are all even. Yes? (That's the only way the question makes sense :))
If so, yes, your solution is correct because the all odd five-digit numbers are disjoint from all even five digit numbers. So the number strings of 5 odd digits, can be added to the number of strings of five even digits, to arrive at $5^5 + 4^5= 4149$ such strings in all. You are indeed correct.
Note: one could find the number of mixed strings (those with at least one odd and at least one even digit) by noting the number of all possible 5-digit strings is $9^5$ and subtracting your total for strictly odd and strictly even, to get $9^5 - (5^5+4^5) = 59049-4149=54900 $ such strings.