Ms. white has 15 students in her first grade class. troy is the line leader for the week, and mackenzie is last because she was the line leader last week. in how many different ways can ms. whites class line up for lunch?

The first place and 15th place are already decided, so we have to find the number of
different ways that the other 13 students can line up, in the places from #2 to #14.

2nd place can be any one of 13 people.  For each of those . . .
3rd place can be any one of 12 people.  For each of those . . .
4th place can be any one of 11 people.  For each of those . . .
.
.
.
13th place can be any one of 2 people.  For each of those . . .
14th place has to be the one student who is left.

Total number of ways that 13 students can line up in places #2 through #14 is

(13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)

That number is called "thirteen factorial".  The number is 6,227,020,800 .

When you write it in math, you write it like this:    13!