CAT 2017 Slot 2 — DILR Question 29
Answer the following question based on the information given below.
A high security research lab requires the researches 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 of the scan order requirements, since it is observed that some employees often get locked-out because they forget the sequence.
The lab has decided to allow a variation in the sequence of scans of 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 & solution
Answer: 11
As per the given information either the entire sequence of scans has to be correct or 3 out of the 5 scans have to be in place. So if we consider TIMRL as the original sequence, each of T, I, M, R and L has to be in the correct sequence which is only possible in 1 way. Now if the sequence of scans is out of place, then at least 2 scans will have to be out of place. Which implies that 3 scans are in place. So 3 out of T, I, M, R and L are in place. Now this is possible in 5C3 or 10 ways. So number of different sequences of scans that are allowed = 1 + 10 = 11 ways.
Answer: 11