1. **Problem statement:** We have 12 members (6 married couples) in a committee. (i) Find the number of ways to seat the 12 members around a round table such that 4 executive subcommittee members do not sit next to each other. (ii) Find the number of ways to arrange them in a row so that all men sit together, all women sit together, and no husband sits next to his wife. --- 2. **Part (i) - Seating around a round table with no two subcommittee members adjacent:** - Total members: 12 - Executive subcommittee members: 4 - Non-subcommittee members: 8 **Step 1:** Total ways to seat 12 people around a round table is $(12-1)! = 11!$ because rotations are considered the same. **Step 2:** We want to seat so that no two of the 4 subcommittee members sit next to each other. **Step 3:** First seat the 8 non-subcommittee members around the table. Number of ways: $(8-1)! = 7!$. **Step 4:** Now, place the 4 subcommittee members in the gaps between the 8 seated members. There are 8 gaps. We must choose 4 gaps out of 8 to seat the subcommittee members so that none are adjacent. Number of ways to choose gaps: $\binom{8}{4}$. **Step 5:** Arrange the 4 subcommittee members in the chosen gaps: $4!$ ways. **Step 6:** Total number of ways: $$7! \times \binom{8}{4} \times 4!$$ **Step 7:** Calculate numeric value: - $7! = 5040$ - $\binom{8}{4} = \frac{8!}{4!4!} = 70$ - $4! = 24$ So total ways = $5040 \times 70 \times 24 = 8,467,200$ --- 3. **Part (ii) - Arranging in a row with all men together, all women together, and no husband next to wife:** - 6 men and 6 women (6 couples) **Step 1:** Treat all men as one block and all women as another block. Number of ways to arrange these two blocks: $2! = 2$ (men first or women first). **Step 2:** Arrange the 6 men within their block: $6!$ ways. **Step 3:** Arrange the 6 women within their block: $6!$ ways. **Step 4:** Now, we must ensure no husband sits next to his wife. Since men and women blocks are separate, the only possible adjacency between husband and wife would be at the boundary between the two blocks. **Step 5:** Check the boundary: - If men block is first, the last man and first woman are adjacent. - If women block is first, the last woman and first man are adjacent. **Step 6:** To avoid husband-wife adjacency at the boundary, the man at the end of the men block and the woman at the start of the women block must not be a couple. **Step 7:** Count the number of arrangements where the boundary pair is a couple and subtract from total. - Total arrangements without restriction: $2 \times 6! \times 6!$ - Number of arrangements where boundary pair is a couple: For men block first: - Fix a couple at boundary: choose which couple to be at boundary: 6 choices. - Arrange remaining 5 men: $5!$ - Arrange women with the wife fixed at first position: arrange remaining 5 women: $5!$ Number of such arrangements: $6 \times 5! \times 5!$ Similarly for women block first, same count. Total boundary couple arrangements: $2 \times 6 \times 5! \times 5!$ **Step 8:** Calculate numeric values: - $6! = 720$ - $5! = 120$ Total arrangements: $2 \times 720 \times 720 = 1,036,800$ Boundary couple arrangements: $2 \times 6 \times 120 \times 120 = 172,800$ **Step 9:** Valid arrangements = Total - Boundary couple arrangements $$1,036,800 - 172,800 = 864,000$$ --- **Final answers:** (i) Number of ways = $8,467,200$ (ii) Number of ways = $864,000$