Link http://www.bigbluehistory.net/bb/Statistics/roster1967-68.html looking at team photo, Cats had 18 varsity players (on scholarship) in 1967-68 season. There were 4 freshmen also on scholarship for a total of 22.
The year before (1966-67) UK had 12 varsity (scholarship) players and 11 freshmen on scholarship for 23 total.
From 1966 through 1971, UK signed 38 players to basketball scholarships for an average of just over 6 per year.
That was commonplace in the 1960's. UK (and others) fielded a varsity team and a freshman team. Most teams signed 5, 6, or 7 players per year. The varsity squad was usually around 15 players with 6 more on the freshman team.
Nowadays, UK has 10 or 11 scholarship players on its roster with a few walk-ons added.