12 chairs are arranged in a row and are numbered 1 to 12. 4 men have to be seated in these chairs so...
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.
Answer/Solution
384
Steps/Work
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=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: B.
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=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: B.