SOLUTION: what is the proper subsets for 4,6,8,9,10

{4,6,8,9,10} has 5 elements, so there are 25 = 32 subsets.  
But that includes the set itself, which not a proper subset,
so there are 31 proper subsets.

 1. { }     <-- the empty subset, also written as Ø, considered proper
 2. {4}
 3. {6}
 4. {8}
 5. {9}
 6. {10}
 7. {4,6}
 8. {4,8}
 9. {4,9}
10. {4,10}
11. {6,8}
12. {6,9}
13. {6,10}
14. {8,9}
15. {8,10}
16. {9,10}
17. {4,6,8}
18. {4,6,9}
19. {4,6,10}
20. {4,8,9}
21. {4,8,10}
22. {4,9,10}
23. {6,8,9}
24. {6,8,10}
25. {6,9,10}
26. {8,9,10}
27. {4,6,8,9}
28. {4,6,8,10}
29. {4,6,9,10}
30. {4,8,9,10}
31. {6,8,9,10}
32. {4,6,8,9,10}  <--this is the only improper subset.

There are 31 proper subsets.

Edwin