SOLUTION: Mr.Santos has set 8 question for Math test. a.How many permutations are there for the 8 questions? b.If the question is the 1st question and the most difficult one is the 8th que

(a)  For 8 objects, there are  8! = 1*2*3*4*5*6*7*8 = 40320 perrmutations.       ANSWER




(b)  Then the easiest and the heaviest questions are fixed at their 1st and 8th positions, respectively, 
     and only 6 questions are really permute.


     For 6 questions, there are 6! = 1*2*3*4*5*6 = 720 permutations, in all.     ANSWER




(c)  8 items have 8! = 8*7*6*5*4*3*2*1 = 40320  permutations.


     But in this problem, we consider not all the permutations.


     We consider only THOSE, where the most difficult question and easiest question are set next to each other.
    in other words, we consider and count only those permutations, where these two questions are neighbor.


    It means, that we consider these two objects as two clued together;

    so we actually consider 7 objects, and this special clued object can exist in two states (ED) or (DE), depending

    which one of the two questions, easy (E) or difficult (D), goes first.


    THEREFORE, the total number of all such possible permutations is 

        2*7! = 2 * (7*6*5*4*3*2*1) = 2 * 5040 = 10080.                            ANSWER




(d)  In part (a) above, I just explained that there are  8! = 40320  permutations for 8 items.


     In part (c), I also explained that there are  2*7! = 2 * 5040 = 10080 permutations, where the most difficult and easiest questions

     go next to each other (= are neighbors).



     Hence, the number of those permutations, where these questions ARE NOT neighbors is the difference

         40320 - 10080 = 30240.                                                  ANSWER