2002 Ford F-150 Truck will not start but cranks over fast.

I drove my truck to work yesterday without event. It was 17 degrees yesterday. When I went to leave work, the vehicle would not start, but would turn over very fast. All the lights worked and the alarm for the key in the ignition also worked.

The battery connections are corroded a little, but since the lights and alarm worked, I am thinking it may be more than just the battery. The battery is 3 year old.

I did notice something else that was unusual. I am not sure it is related but thought it may be connected. Typically when I use the keyless remote to lock the vehicle, the headlights flash to confirm that the vehicle is locked but that function did not work last night.

I have checked the oil and it is clean and full and the coolant level is of adequate mixture as well.

Could it be that the battery has enough to crank the motor that fast but not enough spark to start it?

Hope you can help before I resort to having it towed to a repair facility.