Number of Combinations of Top Three Winners From Group of N

combinatoricspermutations

I'm trying to solve a problem where, in a horse race, there could be any number of horses greater than N.

And I need to find the number of possible combinations of top three winners from N number of horses.

I know that if there are 15 horses, there are 2730 possible combination of top three winners, but I don't know how we arrived at that number.

i.e.

secretariat
war admiral
seattle slew
seabiscuit
man o' war
artax
whirlaway
phar lap
becuphalus
frankel
california chrome
big brown
war emblem
sir barton
affirmed

You could have 

 1. secretariat 
 2. war admiral 
 3. seattle slew

or

 1. secretariat
 2. seattle slew
 3. war admiral

or 

 1. seattle slew
 2. secretariat
 3. war admiral

or 

 1. seattle slew
 2. war admiral
 3. secretariat

or 

 1. war admiral
 2. secretariat
 3. seattle slew

or

 1. war admiral
 2. seattle slew
 3. secretariat

So, there are 6 different combinations, and that's just for the top 3 on this list. What is the equation for getting all possible combinations of top three winners from N number of horses? I'm assuming there is some sort of exponentiation here, but I don't know what it is.

Thanks in advance.

Best Answer

What you are looking for is the number of permutations and is denoted by $^{15}P_3$. This means that you are looking to arrange $3$ items from a list of $15$ unique items.

$$^mP_n=\frac {m!}{(m-n)!}$$ Hence the answer is $^{15}P_3=2730$

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