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.
Select
A) 360 B) 384 C) 432 D) 470 E) 480
First of all, I think the question means that "the chairs numbered 1 AND 8 should be occupied".   So, we have that the chairs numbered 1 AND 8 should be occupied and no two adjacent chairs must be occupied. Notice that chairs #2, 7, and 9 cannot be occupied by any of the men (because no two adjacent chairs must be occupied.).   1-2-3-4-5-6-7-8-9-10-11-12   If the third man occupy chair #3, then the fourth man has 5 options: 5, 6, 10, 11, or 12;   If the third man occupy chair #4, then the fourth man has 4 options: 6, 10, 11, or 12;   If the third man occupy chair #5, then the fourth man has 3 options: 10, 11, or 12;   If the third man occupy chair #6, then the fourth man has 3 options: 10, 11, or 12;   If the third man occupy chair #10, then the fourth man has 1 options: 12.   Total of 5+4+3+3+1= 
<gadget id="calculator">5+4+3+3+1</gadget>
<output>16</output>
16 cases. For each case the 4 men can be arranged in 4! ways, thus the total number of arrangements is 16*4!=384.   Answer: 384.
<result>B</result>
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.