Answer:
112
Step-by-step explanation:
Let A be a subset of S that satisfies such condition.
If 3∈A, then the other three elements of A must be chosen from the set B={1,2,5,6,7,8,9,10} (because 3 cannot be chosen again and 4 can't be alongside 3). B has eight elements, then there are [tex]\binom{8}{3}=56[/tex] ways to select the remaining elements of A (the binomial coefficient counts this). The remaining elements determine A uniquely, then there are 56 subsets A.
If 4∉A, we have to choose the remaining elements of A from the set B={1,2,5,6,7,8,9,10}. B has eight elements, then there are [tex]\binom{8}{3}=56[/tex] ways to select the remaining elements of A. Thus, there are 56 choices for A.
By the sum rule, the total number of subsets is 56+56=112
