Question

In how many ways can a teacher arrange 10 students in the front row if there are 60 total students

Answer 1

The one in the first seat can be any one of 60.
               For each of those . . .
The one in the 2nd seat can be any one of the remaining 59.
               For each of those . . .
The one in the 3rd seat can be any one of the remaining 58.
               For each of those . . .
.
.
.
The one in the 9th seat can be any one of the remaining 52.
               For each of those . . .
The one in the 10th seat can be any one of the remaining 51.

Total number of ways to pick 10 students out of 60
and arrange them in 10 seats =

     (60·59·58·57·56·55·54·53·52·51)  =  273,589,847,200,000,000.
                                        (rounded to the nearest hundred million ways)

Number of ways to pick 10 students out of 60,
no matter how they sit =

           (60·59·58·57·56·55·54·53·52·51) / (10·9·8·7·6·5·4·3·2)

                =  75,394,027,560.  (rounded to the nearest 10 ways)