"Cannot install programs to my HDD" can have several meanings.
If you specify the HDD as the installation location, see rolli59.
If, on the other hand, it is simply installing everything to the SSD, the answer is different. If that's the case, reply to this and I will expound.
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If you installed Win7 with the HDD attached, the odds are that your system is booting from the HDD and will not boot without it attached. The lowest-level parts of the boot sequence may have been written to the HDD. Some people like this, since if the SSD gets corrupted the recovery partition on the HDD can be booted. I personally consider it a pain in the butt. The machine should boot from my OS drive. If I remove all the other drives, it will still boot.
If you care at all about what I wrote above (I will not be offended if you don't), check the partitions on the drives in Disk Manager and see which one has a tiny little 100MB partition hidden at the beginning.