The 1" trichlor tablets will dissolve much faster in a skimmer than in a floater. They also have far less chlorine and CYA per each than a 3" tablet. For every 1ppm of FC delivered by trichlor, expect a .6ppm rise in CYA. 10 1" tablets won't do a whole heck of a lot for a 21K gallon pool. You can weigh 10 tablets and use the pool calculator to find out what that weight in trichlor will add for CYA to your pool. It's not very much. I would imagine less than 1 or 2ppm, if that.
So, when you say "some" chlorine, what exactly is "some"? Some, and fine, and good are words that carry little or no value when it comes to pools. Numbers are much more valuable.
Edit: don't place trichlor in the skimmer. When the pump is off, the trichlor will continue to dissolve creating high concentrations of chlorine in the skimmer and lines to the pump which could damage the equipment. If you're going to use trichlor, always use a floater or a feeder.