TSP - key observation

When planning the next move, you only care about which cities you've visited and where you are now. The exact path taken doesn't matter. If you've visited $\{1, 2, 4\}$ and are at city $4$, the remaining problem is the same whether you came via $1 \to 2 \to 4$ or $2 \to 1 \to 4$.

This overlap is what DP exploits. The bitmask tracks visited cities. Combined with current city, this fully describes the state. That's $n \cdot 2^n$ states instead of $n!$ paths.