Order and Ranking: CAT Previous Year Questions
Q. 1 Instructions
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 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?
Check Solution
Ans: 11
Suppose the initial arrangement is represented by TIMRL.
A maximum of two elements can be in incorrect positions. This condition is met solely when two elements swap their places. The number of ways to choose these two elements from the five available positions is given by the combination formula $^5C_2 = 10$.
Besides these arrangements where two elements are swapped, the original sequence itself is also considered acceptable.
Therefore, the total count of acceptable sequences is the sum of the swapped arrangements and the original arrangement: 10 + 1 = 11.
Q. 2 The lab has decided to allow variations of the original sequence so that input of the scanned sequence of five fingers is allowed to vary from the original sequence by one place for any of the fingers. Thus, for example, if TIMRL is the original sequence, then ITRML is also allowed, but LIMRT is not.
How many different sequences are allowed for any given person’s original scan?
Check Solution
Ans: C
The system accepts an input sequence of five fingers that can deviate from the original sequence by a single positional shift for any finger. This deviation is exclusively accomplished through the swapping of any two adjacent fingers. Let the initial arrangement be TIMRL.
Scenario 1: Exactly one pair of adjacent elements is swapped.
The number of distinct pairs of adjacent elements that can be chosen for swapping is calculated as 5 – 1 = 4.
Scenario 2: Two separate pairs of adjacent elements are swapped.
(i) If the first pair (TI) is swapped, the remaining adjacent pairs available for swapping are (MR) and (RL), resulting in 2 distinct possibilities.
(ii) If the second pair (IM) is swapped, the only remaining adjacent pair available for swapping is (RL), resulting in 1 distinct possibility.
The cumulative count of distinct arrangements achievable through these operations is 4 + 2 + 1 = 7.
When the original, unaltered sequence is also considered, the total number of permissible arrangements becomes 7 + 1 = 8.
Q. 3 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) where at most two scans (out of six) are out of place, 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 for any given person’s original scan?
Check Solution
Ans: 15
There might be two misplaced elements.
The initial arrangement is TIMTRL.
If T is swapped: The possibilities are ITMTRL, MITTRL, RIMTTL, LIMTRT. This results in four scenarios.
If I is swapped: This also leads to four possible arrangements.
If M is swapped: Three distinct arrangements can be formed.
If the second T is swapped: Two unique arrangements emerge.
If R is swapped: Only one configuration is achievable.
The sum of these distinct possibilities is 4 + 4 + 3 + 2 + 1 = 14.
Additionally, the original sequence itself is considered a valid arrangement.
Therefore, the total number of permissible arrangements is 14 + 1 = 15.
Q. 4 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?
Check Solution
Ans: C
1. Given an initial arrangement.
2. Swapping any single pair from the set {LR, RL, LT, TI, IM} results in 5 distinct variations.
3. When LR, LT, and IM are each swapped, the arrangement becomes: RLTLMI.
4. When LR and LT are swapped.
5. When LR and TI are swapped.
6. When LR and IM are swapped.
7. When RL and TI are swapped.
8. When RL and IM are swapped.
9. When LT and IM are swapped.
Combining these possibilities yields a total of 13 distinct arrangements.