Subsets of {1-5}

If the largest element is 5, then the smallest element must be 1. We have two choices for each of 2,3,4 -- they can be "in" the subset or "out" of the subset, so there are $2\times 2 \times 2 = 8$ such subsets.

If the largest element is 4, then the smallest element must be 2, so 1 must be "out". We have two choices for 3 -- it is either "in" the subset or "out" of the subset, so there are $2\times 2 =4$ such subsets.

If the largest element is 3, then the smallest element must also be 3, so the subset must be $\{3\}$ , so there is only 1 possibility.

Adding up these subsets, there are $8+2+1 =11$ possibilities.

If you're interested, these are all of them:

(The 8 with 1 and 5)
{1,5}
{1,2,5}
{1,3,5}
{1,4,5}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{1,2,3,4,5}

(The 2 with 2 and 4)
{2,4}
{2,3,4}

And finally:
{3}