This type of problem is typically handled by doing what was suggested above, coding each possible response dichotomously (0 for no, 1 for yes), then using a base 10 approach to generate a unique code for each possible combination, i.e.:
Obviously if you have 20 sports there's a very large number of possible combinations and the code will be up to 20 digits in length (recent versions of SPSS should be able handle this though). You could also use base 2, i.e:
which will result in a codes of shorter length but which will be harder to read at a glance.
Depending on the size of your dataset you'll find that some combinations are far more common than others and most probably won't exist at all. Run a frequencies on the sportcombo variable and begin labelling your combinations.