When I flip the handle to flush the toilet, the flapper drops back to the hole before only a small aount of water has entered the bowl. In order to get a complete flush, the handle has to be held down for about 10 seconds. I've changed the flapper, but it did not make a difference. My plumber tells me that the toilet is old and needs to be replaced. This is a low-boy water saver about 20 years old. Any ideas would be appreciated.

Perhaps the chain that is connected to the flapper is too short for the flapper to become fully extended? How much water is in the tank? I would check out these simple components first before replacing the entire toilet.

Dabada's on to something. Sounds like your flapper linkage needs adjusting.
With the flapper seated the linkage wants to have 1/4" of play. Too much play and the flapper doesn't get pulled back enough giving you a short flush. Too tight and it lets water seep past the flapper. I adjust the chain by the link and if it needs fine tuning, I bend the flush lever rod a bit untill I have the desired play in the linkage. One more thing about a flapper. you will see where the old flapper hooks onto the base of the white overflow tube. On a older type with no hooks the flapper has a ring that slips down over the overflow tube to the seat. If your tank has hooks, take a sharp knife or scissors and cut the neoprene ring off on the marks provided and hook the flapper on the hooks. Leaving the ring on will interfere with the flush. The water level in your tank should be 3/8 to 1/2" below the top of the overflow tube when the tank's filled. And the little tube wants to run from the ballcock and connect to the white overflow tube.. Good luck, Tom