the swap partition counts but it can be a logical partition inside the extended partition.
I think i now see your problem.
did you tell the installer to put grub in the first partition instead of in the MBR?
the first partition is NTFS and grub will not go on there.
this is what that output tells me:
sda disk is one big windows partiton (78GB)
sdb disk has a windows partition (10GB), then a linux partition(10GB) (mint), then another windows partition (14GB) and last the swap partition (3.8GB).
the real question is how do you really want to setup your system?
and is there anything important (data) on sdb the second disk?
can we move the ntfs data around and re partition the disk?
a linux install only needs about a 4GB / partition and a swap partition that is at most 2x the amount of RAM you have.
if you install multiple linux's you only need one swap partition as they can all use the same one.
well that gives you more to think about.