You can use a H61 board but you would need a BIOS update and in doing that you can't take advantage of all of the new features to the 77 boards like PCI express 3.0 or some other things. If you are planning on overclocking you need the get a Z77 board.
So there is no way around that? The newer h61 wouldn't have the update? Do you have any cheap suggestions for the h77 because I have literally $10 left in the budget.
Nope. H77 is newer than H61 - Ivy Bridge CPUs aren't supported on H61 motherboards. If you want you could use a Radeon 7770 instead of the 6850 - some models go for $100 and then you can upgrade to the H77 motherboard.