Logical Reasoning

Arrangements

Medium

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?

✅ Correct Option: 4

Slide 1/11

Understanding the set :-

This is a basic arrangement set,

Here you have 16 shelves and 12 items to be arranged in the shelves,

Also, we know items are of three types -

a) Biscuits - 5 items

b) Candies - 3 items

c) savouries - 4 items.

we are given that same type of items are clustered together

=> all biscuits , all candies, and all savouries are kept in consecutive shelves.

and, there is atleast 1 and atmost 2 empty shelves between two different type.

Now, we need to find the type and arrangement of all the items using the clues given.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code																K

Item Type	Item code
Biscuits (5)
Candies (3)
savouries (4)

Clue 4 says,

K is to be placed in shelf number 16.

We can directly add this to the table.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code																K

Item Type	Item code
Biscuits (5)
Candies (3)	C
savouries (4)	D, E, F

Clue 3 says,

D, E and F are savouries

Also, According to clue 6,

C is a candy

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code																K

Item Type	Item code
Biscuits (5)	I, L, J
Candies (3)	C
savouries (4)	D, E, F

Clue 5 says,

L and J are items of the same type, while H is an item of a different type

Also, clue 2 says,

I and J are to be placed in consecutively numbered shelves

=> I, L, J are of items of same type

Now, it can't be savouries or candy , as only two and one space is left for candy and savouries respectively.

Therefore, only possibility for I, L, J is biscuits.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code												-	D	E	F	K
Item type												-	S	S	S	S

Item Type	Item code
Biscuits (5)	I, L, J
Candies (3)	C
savouries (4)	D, E, F, K

Clue 3 says,

D, E and F are savouries and are to be placed in consecutively numbered shelves in increasing order after all the biscuits and candies.

=> all savouries are kept after biscuits and candies.

=> Savouries are in shelf number - 13, 14, 15, 16.

And, shelf 12 will be empty, as there is atleast 1 and atmost 2 empty shelves between two different type.

Also, now, we know, K is also a savoury.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code												-	D	E	F	K
Item type												-	S	S	S	S

Item Type	Item code
Biscuits (5)	I, L, J
Candies (3)	C, H
savouries (4)	D, E, F, K

Now according to clue 5,

. L and J are items of the same type, while H is an item of a different type.

=> H is an different item from L, J

=> L is either a candy or savoury.

But, we already found all 4 savoury

Therefore, H will be a candy.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code												-	D	E	F	K
Item type												-	S	S	S	S

Item Type	Item code
Biscuits (5)	I, L, J, A, B
Candies (3)	C, H, G
savouries (4)	D, E, F, K

Now, clue 1 says,

A and B are to be placed in consecutively numbered shelves in increasing order.

=> A and B are of same type

But only Biscuits can accommodate two items, Therefore, A and B will be biscuits.

=> G will be a Candy.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code												-	D	E	F	K
Item type												-	S	S	S	S

Item Type	Item code
Biscuits (5)	I, L, J, A, B
Candies (3)	C, H, G
savouries (4)	D, E, F, K

Now, Clue 6 says,

C is a candy and is to be placed in a shelf preceded by two empty shelves.

Also clue 7 says,

. L is to be placed in a shelf preceded by exactly one empty shelf.

=> we can make two cases :

when candies are first, followed by biscuits and then savouries.

when biscuits are fist, followed by candies and then savouries.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code	-	-	C	H/G	G/H	-	L	A	B	I/J	J/I	-	D	E	F	K
Item type	-	-	C	C	C	-	B	B	B	B	B	-	S	S	S	S

Item Type	Item code
Biscuits (5)	I, L, J, A, B
Candies (3)	C, H, G
savouries (4)	D, E, F, K

CASE - 1 -

First candies will be there,

From clue 6 we can say, 1st will be C

and then H and G in any order.

Then it will be followed by biscuits;

From clue 2 we can say, I & J will be in higher numbered shelf then A & B

=> I & J will be after A & B.

Also, clue 1 says,

A and B are to be placed in consecutively numbered shelves in increasing order.

=> A and B are in increasing order i.e.,

First A will come followed by B.

And, According to clue 7,

L is to be placed in a shelf preceded by exactly one empty shelf.

=> L will come first.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code	-	L	A	B	I/J	J/I	-	-	C	H/G	G/H	-	D	E	F	K
Item type	-	B	B	B	B	B	-	-	C	C	C	-	S	S	S	S

Item Type	Item code
Biscuits (5)	I, L, J, A, B
Candies (3)	C, H, G
savouries (4)	D, E, F, K

Now, CASE -2 :

First biscuits will come;

(In the same order as in previous case.)

followed by Candies.

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code	-	-	C	H/G	G/H	-	L	A	B	I/J	J/I	-	D	E	F	K
Item type	-	-	C	C	C	-	B	B	B	B	B	-	S	S	S	S

Shelf No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Item Code	-	L	A	B	I/J	J/I	-	-	C	H/G	G/H	-	D	E	F	K
Item type	-	B	B	B	B	B	-	-	C	C	C	-	S	S	S	S

Item Type	Item code
Biscuits (5)	I, L, J, A, B
Candies (3)	C, H, G
savouries (4)	D, E, F, K

The number of arrangements for the first case = 2*2=4

The number of arrangements for the second case = 2*2=4

The total number of arrangements = 4+4=8.

Question 14

Which of the following items is not a type of biscuit?

Question 15

Which of the following can represent the numbers of the empty shelves in a possible arrangement?

Question 16

Which of the following statements is necessarily true?

In how many different ways can the items be arranged on the shelves?

Solution

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