SOLUTION: How many permutations of the letters {A, B, C , D, E , F , G } are there, such that 1 A precedes B 2 A precedes B, and C precedes D 3 A precedes B, and B precedes C 4 C,D,E a

That's the number of permutations of these 6 things AB, C, D, E, F, G
taken all 6 at a time.

That's the number of permutations of these 5 things AB, CD, E, F, G
taken all 5 at a time.

That's the number of permutations of these 5 things ABC, D, E, F, G
taken all 5 at a time.

That's the number of permutations of these 5 things A, B, CDE, F, G
taken all 5 at a time.

That's the number of permutations of these 5 things AB, CD, E, F, G
taken all 5 at a time.

You can calculate all these numbers of permutations yourself.  They're just
factorials.

Edwin

     In English, "A precedes B" means that A goes/comes before B, but it is not necessary 
     that B goes/comes immediately after A: it may happen that there is / (there are) other letter / (letters) 
     after A, but before B.  In other words, "A precedes B" means that A goes/comes before B,
     but A and B are not necessary adjacent (= not necessary appear altogether).

     Based on this understanding, there are 7! = 7*6*5*4*3*2*1 = 5040 permutations of 7 letters
     {A, B, C , D, E , F , G}; of them, in exactly half cases, A precedes B. In other half, 
     in opposite, B precedes A.  To convince yourself that it is true, take any permutation
     where A precedes B and make swap.

     So, the ANSWER to (1) is   =  = 2520.

     As we just learned from (a), there are   =  = 2520 permutations 
     of 7 letters {A, B, C , D, E , F , G}, such that A precedes B.

     Of them, in half cases C precedes D; in other half cases D precedes C.

     So, the ANSWER for (2) is   = 1260.  Or, which is the same,  = .

     With any permutation of 7 letters  {A, B, C , D, E , F , G }, there are 3!-1 = 6-1 = 5 other permutations
     where letters A, B and C are swapped in their positions.

     It gives you an idea, that with any favorable permutation, where A precedes B and B precedes C, there are 5 other
     permutations with swapped A, B and C in their positions.

     So, the answer to (c) is   =  =  =  840.

     In this part, formulation in the post is unclear (= does not sound harmonically *) and needs to be clarified.

     May be, you want to ask "A,B appear together in the same order, as do C,D"

     Then it is clear, but it is DIFFERENT from your original formulation.

     Having unclear formulation, I prefer do not touch this part.