Question 1203598
.
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 appear together in this order
5 A,B {{{highlight(cross(are))}}} appear together in this order, as do C,D
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        My reading and understanding is different from that of Edwin,

        so my solutions and my answers are different.



(1)    A precedes B


<pre>
     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 <U>ANSWER</U> to (1) is  {{{7!/2}}} = {{{5040/2}}} = 2520.
</pre>


(2)  &nbsp;&nbsp;A precedes B, and C precedes D


<pre>
     As we just learned from (a), there are  {{{7!/2}}} = {{{5040/2}}} = 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 <U>ANSWER</U> for (2) is  {{{2520/2}}} = 1260.  Or, which is the same, {{{7!/(2*2)}}} = {{{7!/4}}}.
</pre>


(3)  &nbsp;&nbsp;A precedes B, and B precedes C


<pre>
     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  {{{7!/3!}}} = {{{7!/6}}} = {{{5040/6}}} =  840.
</pre>


(4) &nbsp;&nbsp;C,D,E appear together in this order.


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;For this part, I agree with Edwin.



(5)  &nbsp;&nbsp;A,B appear together in this order, as do C , D


<pre>
     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.
</pre>

Solved (to that extent as possible).


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*) &nbsp;&nbsp;In &nbsp;Math, &nbsp;if a problem does not sound harmonically, &nbsp;it is a sure sign that it is formulated incorrectly.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Why ? - Because harmony is the hidden meaning of &nbsp;Math.