Any single number covers sums $1$ through $5.$ If a consecutive block sums to $n,$ the remaining numbers form a consecutive block summing to $15 - n,$ so sums $10$ through $14$ are automatically covered as well. Thus an arrangement is bad only if it fails to produce $6$ or $7.$

If $6$ cannot be formed, rotate and reflect so the order is $1bc5e.$ The pair $\{b,c\}$ cannot be $\{2,3\}$ or $\{2,4\},$ so $e=2;$ avoiding the block $213$ then forces $14352.$ If $7$ cannot be formed, write the order as $2bc5e.$ Now $\{b,c\}$ cannot be $\{3,4\}$ or $\{1,4\},$ so $e=4,$ and avoiding $421$ forces $23154.$

These are the only two bad arrangements up to rotation and reflection.

Thus, the correct answer is B.