#### 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 Question : The lab has decided to allow a variation in the sequence of scans of the five fingers so that at most two scans (Out of five) are Out of place. For example, if the original sequence is Thumb Cr), index finger (I), middle finger (M), ring finger CR) and little finger (L) then TLMRI is also allowed, but TMRU is not. How many different sequences of scans are allowed for any given persons original scan? Option: 1 11Option: 2 -Option: 3 -Option: 4 -

The question asks us to find how many different sequences of scans are allowed for any given person's original scan, given that the lab has decided to allow a variation in the sequence of scans of the five fingers so that at most two scans (out of five) are out of place.

We can use the following cases to solve the problem:

• Case 1: All the scans are in the correct order.
• Case 2: Exactly two scans are out of place.

For Case 1: There is only one way to do this, since the order of exactly one out of the five scans cannot be changed.

For Case 2: We can choose any two of the five scans that can be interchanged in (25?) =10 ways.

Therefore, the total number of acceptable sequences is 1+10=11?

Here is a table that summarizes the two cases:

 Case Number of sequences All scans in the correct order 1 Exactly two scans out of place 10 Total 11