Question

There is a class of 20, and we are going to pick a team of three people at random, and we want to know:

how many different possible three-person teams could we pick? Another way to say that is: how many different combinations of 3 can be taken from a set of 20?

Answer 1

If n is the size of the larger collection, and r is the number of elements that will be selected, then the number of combinations is given by

In the question just posed, n = 20, r = 3, and n – r = 17.  Therefore

To simplify this, consider that:

20 = (20)(19)(18)(17)

That neat little trick allow us to enormously simplify the combinations formula:

= 1140

Answer 2

If n is the size of the larger collection, and r is the number of elements that will be selected, then the number of combinations is given by

# of combinations = $\frac{n!}{r!(n-r)!}$

In the question just posed, n = 20, r = 3, and n – r = 17.  Therefore,

# of combinations = $\frac{20!}{3(17)!}$ 

To simplify this, consider that:

20! = (20)(19)(18)(17)(the product of all the numbers less than 17)

 

Or, in other words,

20! = (20)(19)(18)(17!)

 

That neat little trick allow us to enormously simplify the combinations formula:

# of combinations =  $\frac{(20)(19)(18)(17)}{3!(17)!}$ $\frac{(20)(19)(18)}{(3) (2) (1)}$ $\frac{(20)(19)(18)}{(3)(2)(1)}$ = 1140