A high security research lab requires the researchers to set a pass key sequence based on the scan of the five fingers of their left hands. When an employee first joins the lab, her fingers are scanned in an order of her choice, and then when she wants to re-enter the facility, she has to scan the five fingers in the same sequence.
The lab authorities are considering some relaxations of the scan order requirements, since it is observed that some employees often get locked-out because they forget the sequence.
The lab has now decided to require six scans in the pass key sequence, where exactly one finger is scanned twice, and the other fingers are scanned exactly once, which can be done in any order. For example, a possible sequence is TIMTRL.
Suppose the lab allows a variation of the original sequence (of six inputs) so that input in the form of scanned sequence of six fingers is allowed to vary from the original sequence by one place for any of the fingers, as long as the finger originally scanned twice is scanned twice and other fingers are scanned once.
How many different sequences of scans are allowed if the original scan sequence is LRLTIM?
1. If original sequence is given.
2. If either of LR, RL, LT, TI, IM is interchanged => 5 ways.
3. If LR and LT and IM interchanged. The sequence will look like: RLTLMI
4. If LR and LT are interchanged.
5. If LR and TI are interchanged.
6. If LR and IM are interchanged.
7. If RL and TI are interchanged.
8. If RL and IM are interchanged.
9. If LT and IM are interchanged.
Total 13 ways possible.