CAT 2019 Slot 1 — DILR Question 1
Answer the following question based on the information given below.
A supermarket has to place 12 items (coded A to L) in shelves numbered 1 to 16. Five of these items are types of biscuits, three are types of candies and the rest are types of savouries. Only one item can be kept in a shelf. Items are to be placed such that all items of same type are clustered together with no empty shelf between items of the same type and at least one empty shelf between two different types of items. At most two empty shelves can have consecutive numbers.
The following additional facts are known.
- A and B are to be placed in consecutively numbered shelves in increasing order.
- I and J are to be placed in consecutively numbered shelves both higher numbered than the shelves in which A and B are kept.
- D, E and F are savouries and are to be placed in consecutively numbered shelves in increasing order after all the biscuits and candies.
- K is to be placed in shelf number 16.
- L and J are items of the same type, while H is an item of a different type.
- C is a candy and is to be placed in a shelf preceded by two empty shelves.
- L is to be placed in a shelf preceded by exactly one empty shelf.
In how many different ways can the items be arranged on the shelves?
Answer & solution
- A
1
8
- C
2
- D
4
There are 5 types of biscuits, 3 types of candies and 4 types of savouries. Among 16 shelves, there are 4 empty shelves.
It is given that all items of same type are clustered together with no empty shelf between items of the same type.
From (3) and (4), it can be concluded that D, E, F and K are savouries.
From (2) and (5), L, I and J are of one type and H is the other type. Therefore from (6), as C is a candy, L, I J must be types of biscuits and H is a type of candy. Now using (1), we can conclude that A and B are of one type but not candies as there are only 3 types of candies.
Therefore,
Biscuits: A, B, I, J, L Candies: C, H, G Savouries: D, E, F, K
From (3), (4), (6) and (7), there shelf number 12 must be an empty shelf. Also, D, E, F and K are placed in shelves numbered 13, 14, 15 and 16 respectively.
Now from (1), (2) and (7), the sequence (from left to right) in which biscuits are kept is:
(Empty shelf), L, A, B, (I/J), (J/I).
From (6), the candies must be in the following order: (Empty shelf), (Empty shelf), C, (H/G), (G/H)
Thus, we have
âââââââIn each case, J and I can be arranged in 2 ways and G and H can be arranged among them in 2 ways. Thus, 2 × 2 = 4 ways.
Total number of ways the items can be arranged on the shelves = 4 + 4 = 8
Hence, option (b).