SOLUTION: Six students are to line up for a photograph. a) In how many ways can the 6 students standing in a line be arranged? b) In how many ways can the 6 students standing in a line

(a)  in 6! = 6*5*4*3*2*1 = 720 different line arrangements.

     The number of permutations of n objects is n! = 1*2*3* . . . *n.

     The number of permutations of 6 objects is 6! = 1*2*3* . . . *6 = 720.


     Or use this mantra

       - any of 6 students can stay in 1st position (giving 6 options)
       - any of 5 remaining students can stay  in 2nd position (giving 5 options)
       - any of 4 remaining students can stay  in 3rd position (giving 4 options)
                . . . and so on . . . 
       - any of 2 remaining students can stay  in 5th position (giving 2 options)
       -          remaining last student stays in 6th position (giving only 1 options).


      Then you multiply the number of options and get 6!.




(b)  5! = 120, because the first position is just occupied,

         and only 5 remaining students can permutate in 5 remaining positions.
  

        

(c)  4! =  24, because the 1st position and the 6th position are just occupied,

         and only 4 remaining students can permutate in 4 remaining positions.