F.4 maths probability

7 places in a row is like this: _ _ _ _ _ _ _

Question:
"What is the meaning of stand together and next to?"

Answer:
The question requires that
★ A and B are adjacent to each other (that is, (A, B) or (B, A)), and
★ C cannot be next this group of (A, B) or (B, A).

Solution:
First ignore the relative position of A and B, that is, temporarily ignore (A, B) vs (B, A)
There are 6 cases in total:
(1) A B _ _ _ _ _ there are 4 places for C
(2) _ A B _ _ _ _ there are 3 places for C
(3) _ _ A B _ _ _ there are 3 places for C
(4) _ _ _ A B _ _ there are 3 places for C
(5) _ _ _ _ A B _ there are 3 places for C
(6) _ _ _ _ _ A B there are 4 places for C

Therefore, the total number of satisfied arrangements is
2 × (4 + 3 + 3 + 3 + 3 + 4) × 4! = 960

Note:
2 is for (A, B) and (B, A)
4! is for the remaing 4 persons other than A, B and C.

The total number of all possible arrangements is 7! = 5040

Thus, the required probability is
960/5040 = 4/21