# Pick the answer not the lock

Algebra Level pending

. Let K1, K2, K3, K4, K5 be 5 distinguishable keys, and let D1, D2, D3, D4, D5 be 5 distinguishable doors. For 1 ≤ i ≤ 5, key Ki opens doors Di and Di+1 (where D6 = D1) and can only be used once. The keys and doors are placed in some order along a hallway. Key$ha walks into the hallway, picks a key and opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked. In how many such ways can the keys and doors be ordered if Key$ha can open all the doors? Clarifications: • The doors and keys are in series. In other words, the doors aren’t lined up along the side of the hallway. They are blocking Key\$has path to the end, and the only way she can get past them is by getting the appropriate keys along the hallway. • The doors and keys appear consecutively along the hallway. For example, she might find K1D1K2D2K3D3K4D4K5D5 down the hallway in that order. Also, by “she never obtains a key before all the doors in front of it are unlocked, we mean that she cannot obtain a key before all the doors appearing before the key are unlocked. In essence, it merely states that locked doors cannot be passed. • The doors and keys do not need to alternate down the hallway.

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