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 to 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

• Given there are 12 numbered chairs, such that chairs numbered 1 to 8 should be occupied.
  _X_ , 2,3,4,5,6,7,_X_,9,10,11,12.
  
  The various combinations of chairs that ensure that no two men are sitting together are listed.
  (1,3,5,__). The fourth chair can be 5,6,10,11 or 12, hence 5 ways.
  (1,4,8,__), The fourth chair can be 6,10,11 or 12 hence 4 ways.
  (1,5,8,__), the fourth chair can be 10,11 or 12 hence 3 ways.
  (1,6,8,__), the fourth chair can be 10,11 or 12 hence 3 ways.
  (1,8,10,12) is also one of the combinations.
  
  Hence, 16 such combinations exist.
  In case of each these combinations we can make the four men inter arrange in 4! ways.
  
  Hence, the required result =16×4!= 384
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