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 (T), index finger (I), middle finger (M), ring finger(R) and little finger (L) then TLMRI is also allowed, but TMRLI is not.
How many different sequences of scans are allowed for any given person's original scan?___

Answer 1

According to question, at most two scans are out of place. So, there can be 0, 1, or 2 misplaced.
Case 1: Zero out of place. In this case none of the finger is misplaced, meaning this is an original sequence. Number of sequence possible = 1
Case 2: One out of place. This case is not possible as we need two placed to be misplaced. Number of sequence possible = 0
Case 3: Two out of place. Three will be at their original place and two will be misplaced.
For two selected, two arrangements are possible but one will lead to the original sequence, hence only one can will be of our need. Selecting two out of five i.e. ${}^5{C}_2=10$
Therefore, total number of arrangements possible = 10 + 0 + 1 = 11