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?
Which of the following items is not a type of biscuit?
Which of the following can represent the numbers of the empty shelves in a possible arrangement?
Which of the following statements is necessarily true?
- There are two empty shelves between the biscuits and the candies.
- All biscuits are kept before candies.
- There are at least four shelves between items
- All candies are kept before biscuits.
There are 12 items such that, 5 of them are biscuits, 3 of them are candies and 4 of them are savories.
It is given that K is in the shelf numbered 16.
D, E and F are placed in consecutively numbered shelves in the ascending order. They are placed after biscuits and candies. From this, it can be inferred that D, E and F are savories and they are placed at the end. Since K is in the last shelf, K is also a savory and all items of the same type are placed in consecutively numbered shelves, D, E, F and K will be in shelves numbered 13, 14, 15 and 16 respectively.
Now L and J are same type of items and since I and J are in consecutively numbered shelves, L, I and J are either biscuits or candies. It is given that C is a candy. If L, I and J are candies, then there would be a total of four candies. But it is not possible because there are only three candies. Therefore, L, I and J are biscuits. H is different from L. Therefore, H has to be a candy. Both A and B are in consecutively numbered shelves. This implies that they are of the same type. If A and B are candies, then the total number of candies will be 4. But this is not possible because there are only three candies. Therefore, A and B are biscuits.
We have two cases here.
Case 1: Candies are placed after biscuits
It is given that L is placed after exactly one empty shelf. Since biscuits are placed before candies, L is to be placed in shelf 2. Hence, the shelves numbered 2, 3, 4, 5 and 6 will have biscuits in them. I and J are in shelves numbered higher than those of A and B. Therefore. A and B will be in shelves 3 and 4 respectively. I and J will be in shelves 5 and 6. not necessarily in that order.
C is placed in a shelf preceded by two empty shelves. Therefore. C will be in shelf 9. Both H and G will be in shelves 10 and 11 in no particular order. Shelf numbered 12 will be empty.
The shelves numbered 1, 7,8 and 12 are empty.
Case 2: Biscuits are placed after candies
C is placed in a shelf preceded by two empty shelves. Since candies are placed before biscuits, C will be in shelf 3. Both H and G will be in shelves 4 and 5, not necessarily in that order. Shelf 6 will be empty.
It is given that L is placed after exactly one empty shelf. Therefore. L will be in shelf 7. Hence, the shelves numbered 7, 8, 9,10 and 11 will have biscuits in them.
I and J are in shelves numbered higher than those of A and B. Therefore. A and B will be in shelves 8 and 9 respectively. I and J will be in shelves 10 and 11. not necessarily in that order. Shelf numbered 12 will be empty.
The arrangement will be as follows:
The shelves numbered 1. 2, 6 and 12 are empty.
The items can be arranged in 4 ways in the first case and 4 ways in the second case. Therefore, they can be arranged in a total of 8 ways. Answer: (8)
G is not a biscuit.
Shelves numbered 1, 2, 6 and 12 can be empty.
Answer: (1,2, 6. 12)
There are at least four shelves in between B and C in both the cases. This is true.
Answer: (There are at least four shelves between items B and C.)
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