We say that a set of integers is sparse if any two elements in the set differ by at least 3. Find the number of sparse subsets of 1, 2, 3, ... , 12, 13, 14 (Note that the empty set is also sparse.)
+23246 I do not know what the answer is, but, in listing some sets, I get more than 56 sets when counting only those that include the number 1.
+118609 We say that a set of integers is sparse if any two elements in the set differ by at least 3. Find the number of sparse subsets of 1, 2, 3, ... , 12, 13, 14 (Note that the empty set is also sparse.)
I get 51 that are NOT sparse
The number of subsets altogther is
1+14+14C2+14C3+14C4+14C5+14C6+14C7+14C8+14C9+14C10+14C11+14C12+14C13+14C14
=2+28+2(14C2+14C3+14C4+14C5+14C6)+14C7
=30+2(14C2+14C3+14C4+14C5+14C6)+14C7
=30+2(91+364+1001+2002+3003)+3432
=3462+2(6461)
=16384
So the number of subsets that are sparse is 16384-51 = 16333
