Hello Akader
I agree with @davidilles, the calculations are correct. Let me add one more thought, which has to do with real-world considerations. I know this wasn’t part of the actual exercise, but I believe it is beneficial to consider when applying this in practice.
If you have 250 hosts in a subnet, you must also consider future growth. Using a subnet size of 256 works, but you will quickly find that there will be an extra printer to be added, or another user will need connection, quickly reaching the limit. A rule of thumb is to use the 20% rule, where 20% of your subnet range remains initially unused.
This of course assumes that the original IP address range you are provided with has room to create larger subnets. In any case, it is a parameter that must be considered.
Finally, at the end, you mention the free space with a network address of 10.0.5.164. ΅What was the original IP address space provided? If it was 10.0.0.0/21 that would give us a range from 10.0.0.0-10.0.7.255 which would work for the subnetting you created. So if you want to create new subnets, you would indeed begin with 10.0.5.164. Just keep in mind that if you do the calculations, you’ll find that the largest address space you can get for that network address is a /31. If you need larger subnet ranges, you would probably move on to 10.0.5.192/26 as the next network address space or even 10.0.6.0/24. If you do the calculations, you will see why this is.
This is not a problem with your calculations, you’ve worked them out correctly, it’s just the nature of the result. I hope this has given a bit more insight into the process, and not added more confusion… 
I hope this has been helpful!
Laz