basic aptitude

Question

12 chairs are arranged in a row and are numbered 1 to 12 4 men have to be seated in these chairs so that the chairs numbered 1 and 8 should be occupied and no two men occupy adjacent chairs. Find the number of ways the task can be done.

	A.	360
	B.	384
	C.	432
	D.	470

Answer 1

The arrangement is like this:

_  _  _  _  _  _  _  _  _  _  _  _

In the above arrangement, seat number 1 and 8 is always filled

X  _  _  _  _  _  _  X  _  _  _  _

In seat number 1 and 8 we have to select from 4 Men so we have order dependent and no repetition arrangement i.e. 4P2 = 12

 Now no two men should seat on adjacent chairs.

Nobody can seat in chair number 2, 7, 9.

Chairs which have only 1 valid neighbour are 4 chairs (chair number 3, 6, 10, 12).

Chairs which have 2 valid neighbours are 3 chairs (chair number 4, 5, 11).

Hence for 1 neighbour, we get 5 available seats (eg. if seat number 3 gets filled we get options as seat number 5,6,10,11 or 12)

and for 2 neighbours, we get 4 available seats (eg. if seat number 4 gets filled we get options as seat number 6,10,11 or 12)

So we have (4*5) + (3*4) = 32

Finally 32 * 12 = 384

Option B.

Answer is open to any suggestions

Comments

Chairs which have 2 valid neighbours are 3 chairs (chair number 4, 5, 11). can you explain  this one

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Consider chair number 4, we can count chair number 3 and chair number 5 both to be excluded from counting as they are not neighbours to 1 or 8.

But for chair number 3, we cannot count chair number 2 as a neighbour of 3 because it is already excluded due to neighbour of 1.

If u find this neighbour theory difficult, then consider it like this

after filling 1 and 8 numbered seat we are left with 10-3 = 7 seats (because 2,6 and 7 numbered seats are not available to fill)

Here we cannot simply do 7*6 because for filling chairs in the 2nd round we have to make sure that no two neighbouring chairs are filled.

For 3  2-neighboured chairs (numbered 4,5 and 11) we have to exclude 2 chairs from 6 so we get 3*4

For 4  1-neighboured chairs (numbered 3,6,10 and 12) we have to exclude 1 chair from 6 so we get 4*5

hence instead of 7 * 6 we have to use (3*4)+(4*5)

Hope this helps

Answer 2

B..

16*4!=384

Comments

explain ?