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.
Q.1 In how many different ways can the items be arranged on the shelves?
- 4
- 2
- 8
- 1
Explanation
Answer: 3
Certainly! Here’s a rephrased version of the provided solution:
The given configuration involves a total of 5 biscuits and 3 candies. Hence, the remaining items, savouries, amount to 12 – 5 – 3 = 4. Considering the arrangement constraints, it’s evident that D, E, F, and K are the 4 savouries occupying shelves numbered 13, 14, 15, and 16, with no empty shelves between items of the same type.
Building on this information, we can infer that in the sequence I, J, and L, L occupies the lowest-numbered shelf among the three. Further, according to point VI, item C is a candy. Consequently, items I, J, and L must be biscuits, as there are only 3 candies in total.
Expanding on the deductions, we find that H, not being a biscuit, must be a candy. This leaves A and B as the remaining biscuits. Therefore, we can now layout the items as follows:
Case 1: All biscuits are placed after candies.
Case 2: All candies are placed after biscuits.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Case 1 C H/G G/H L A B I/J J/I D E F K
Case 2 L A B I/J J/I C H/G G/H D E F K
In total, there are 8 possible cases. The provided information allows us to answer the questions regarding the arrangement:
- There are two empty shelves between the biscuits and the candies.
- All candies are placed before biscuits.
- All biscuits are placed before candies.
- There are at least four shelves between items B and C.
Q.2 Which of the following items is not a type of biscuit?
Explanation
The given configuration involves a total of 5 biscuits and 3 candies. Hence, the remaining items, savouries, amount to 12 – 5 – 3 = 4. Considering the arrangement constraints, it’s evident that D, E, F, and K are the 4 savouries occupying shelves numbered 13, 14, 15, and 16, with no empty shelves between items of the same type.
Building on this information, we can infer that in the sequence I, J, and L, L occupies the lowest-numbered shelf among the three. Further, according to point VI, item C is a candy. Consequently, items I, J, and L must be biscuits, as there are only 3 candies in total.
Expanding on the deductions, we find that H, not being a biscuit, must be a candy. This leaves A and B as the remaining biscuits. Therefore, we can now layout the items as follows:
Case 1: All biscuits are placed after candies.
Case 2: All candies are placed after biscuits.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Case 1 C H/G G/H L A B I/J J/I D E F K
Case 2 L A B I/J J/I C H/G G/H D E F K
In total, there are 8 possible cases. The provided information allows us to answer the questions regarding the arrangement:
- There are two empty shelves between the biscuits and the candies.
- All candies are placed before biscuits.
- All biscuits are placed before candies.
- There are at least four shelves between items B and C.
Q.3 Which of the following can represent the numbers of the empty shelves in a possible arrangement?
- 1,7,11,12
- 1,5,6,12
- 1,2,6,12
- 1,2,8,12
explanation
Answer: 3
The given configuration involves a total of 5 biscuits and 3 candies. Hence, the remaining items, savouries, amount to 12 – 5 – 3 = 4. Considering the arrangement constraints, it’s evident that D, E, F, and K are the 4 savouries occupying shelves numbered 13, 14, 15, and 16, with no empty shelves between items of the same type.
Building on this information, we can infer that in the sequence I, J, and L, L occupies the lowest-numbered shelf among the three. Further, according to point VI, item C is a candy. Consequently, items I, J, and L must be biscuits, as there are only 3 candies in total.
Expanding on the deductions, we find that H, not being a biscuit, must be a candy. This leaves A and B as the remaining biscuits. Therefore, we can now layout the items as follows:
Case 1: All biscuits are placed after candies.
Case 2: All candies are placed after biscuits.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Case 1 C H/G G/H L A B I/J J/I D E F K
Case 2 L A B I/J J/I C H/G G/H D E F K
In total, there are 8 possible cases. The provided information allows us to answer the questions regarding the arrangement:
- There are two empty shelves between the biscuits and the candies.
- All candies are placed before biscuits.
- All biscuits are placed before candies.
- There are at least four shelves between items B and C.
Q.4 Which of the following statements is necessarily true?
- There are two empty shelves between the biscuits and the candies
- All candies are kept before biscuits
- All biscuits are kept before candies.
- There are at least four shelves between items B and C.
explanation
Anwser 4
The given configuration involves a total of 5 biscuits and 3 candies. Hence, the remaining items, savouries, amount to 12 – 5 – 3 = 4. Considering the arrangement constraints, it’s evident that D, E, F, and K are the 4 savouries occupying shelves numbered 13, 14, 15, and 16, with no empty shelves between items of the same type.
Building on this information, we can infer that in the sequence I, J, and L, L occupies the lowest-numbered shelf among the three. Further, according to point VI, item C is a candy. Consequently, items I, J, and L must be biscuits, as there are only 3 candies in total.
Expanding on the deductions, we find that H, not being a biscuit, must be a candy. This leaves A and B as the remaining biscuits. Therefore, we can now layout the items as follows:
Case 1: All biscuits are placed after candies.
Case 2: All candies are placed after biscuits.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Case 1 C H/G G/H L A B I/J J/I D E F K
Case 2 L A B I/J J/I C H/G G/H D E F K
In total, there are 8 possible cases. The provided information allows us to answer the questions regarding the arrangement:
- There are two empty shelves between the biscuits and the candies.
- All candies are placed before biscuits.
- All biscuits are placed before candies.
- There are at least four shelves between items B and C.
