7人分前后两排（3前4后）且满足A、B相邻A、C不相邻的排列数求解咨询（附错误解题思路）

Hey everyone, I'm stuck on this permutation problem and my current approach is giving an incorrect result. Let me lay out the problem and my thought process below—hoping someone can spot where I went wrong!

Seven people are standing in two rows, with three people in the front row and four in the back row. Among them, A and B must stand next to each other, while A and C must stand separately. How many different arrangements are there?

Here's how I tried to solve it, but the answer isn't right:

First, since A and B have to be next to each other, I grouped them into a single "combined entity". That leaves us with 6 total entities to arrange: 5 individual people plus the A-B unit. The setup has 3 spots in the front row and 4 in the back.

• Step 1: Choose 3 entities out of the 6 to place in the front row. I calculated this as $\binom{6}{3}$ ways.
• Step 2: Arrange the 3 selected entities in the front row: $3!$ possible arrangements.
• Step 3: Arrange the remaining 3 entities (including the A-B unit) in the back row: $3!$ ways. Plus, the A-B unit itself has 2 internal orderings (A on the left of B, or vice versa).
• Step 4: I tried to subtract the invalid arrangements where A and C are standing together. I split this into two cases:
  • Case 1: A and C are in the front row. I thought there were $\binom{4}{1}$ choices for the third entity in the front, and $\binom{3}{2} \times 2$ ways to arrange the back row.
  • Case 2: A and C are in the back row. I calculated $\binom{4}{2} \times 2$ choices for the other two people in the back, and $\binom{3}{1}$ ways to pick one person for the front row.

Putting it all together, my formula for the total number of valid arrangements was:
$$N = \binom{6}{3} \times 3! \times 3! \times 2 - \binom{4}{1} \times \binom{3}{2} \times 2 - \binom{4}{2} \times \binom{3}{1}$$

When I did the calculations:
$$N = 20 \times 6 \times 6 \times 2 - 4 \times 3 \times 2 - 6 \times 3$$
$$N = 1440 - 24 - 18$$
$$N = 1408 - 18$$
$$N = 1390$$

I ended up with 1390, but I know this is incorrect. Can anyone help me identify the mistakes in my reasoning?

备注：内容来源于stack exchange，提问作者csn899