This question, I suspect, has formed the basis of many successful bar bets, but there is no trick to it. Of course, the word ‘racecourse’ contains all of the specified letters, so on-the-ball respondents could argue that the correct answer is ‘all of them’, in which case you might like ask them to name all 59 racecourses on mainland Britain! Seriously though, there are, legitimately, six racecourse in Britain whose names contain all of the letters A, E, R and C without taking the word ‘racecourse’ into account.
Interestingly, all six racecourses are in England, specifically Northern England, with two in Cumbria, two in North Yorkshire, one in West Yorkshire and one in South Yorkshire. If you wish to prolong the agony of your chosen ‘victim’ with further clues, just one of them is exclusively a National Hunt racecourse and just one of them, but not the same one, is right-handed; two are exclusively Flat courses and three operate under both codes. Alphabetically, which you may find the easiest way to remember them, the racecourses in question are Carlisle, which is the only right-handed course, Cartmel, which is the only exclusively National Hunt course, Catterick, Doncaster, Pontefract and Redcar.
As a footnote, if you need a ‘double or quits’ question of similar ilk, you might like to know that there are four British racecourses whose names contain none of the letters A, E, R and C. Again, in alphabetical order, they are Goodwood, Huntingdon, Ludlow and Plumpton. If you want to offer a ‘teaser’ or two about this quartet, Goodwood stages only Flat racing, while the other three stage only National Hunt racing; Plumpton is the only one that is left-handed.