SOLUTION: 15 gymnasts are competing in a national competition and will be ranked from 1 to 5 top five gymnasts will then move on to the world competition how many different ways can the to

Order matters in this problem because of the rankings #1 through #5.

There are 15 ways to choose gymnast #1.

For each of those 15 ways to choose gymnast #1, there remain 14 ways
to choose gymnast #2.

That's 15*14 ways to choose gymnasts #1 and #2.

For each of those 15*14 ways to choose gymnasts #1 and #2, there remain
13 ways to choose gymnast #3.

That's 15*14*13 ways to choose gymnasts #1,#2, and #3.

For each of those 15*14*13 ways to choose gymnasts #1,#2, and #3, there 
remain 12 ways to choose gymnast #4.

That's 15*14*13*12 ways to choose gymnasts #1,#2,#3 and #4.

For each of those 15*14*13*12 ways to choose gymnasts #1,#2,#3 and #4,
there remain 11 ways to choose gymnast #5.

That's 15*14*13*12*11 ways to choose gymnasts #1,#2,#3,#4 and #5.

So the answer is 15*14*13*12*11 = 360360.

That is sometimes called "15 Position 5" or "15P5" or "P(15,5)" or 
"The number of permutations of 15 things taken 5 at a time. 

The general formula for the number of n things taken r at a time
is .

So 

Edwin