Question 971427
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 top five gymnasts be ranked?
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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 {{{nPr = n!/(n-r)!}}}.

So {{{15P5=15!/(15-5)!=15!/10!=(15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/(10*9*8*7*6*5*4*3*2*1)=(15*14*13*12*11*cross(10*9*8*7*6*5*4*3*2*1))/(cross(10*9*8*7*6*5*4*3*2*1))=15*14*13*12*11= 360360}}}

Edwin</pre>