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If I have 5 different varieties of flowers:

carnations, roses, mums, marigolds and lilies.

And I put 3 different varieties of flowers in a vase

For example, a vase might contain roses, carnations and lilies

How many different combinations can be made from the 5 varieties of flowers?

carnations, roses, mums, marigolds and lilies.

And I put 3 different varieties of flowers in a vase

For example, a vase might contain roses, carnations and lilies

How many different combinations can be made from the 5 varieties of flowers?

Cx,y = x!

_________

y! * (x - y)!

In you example:

C5,3 = 5! = 120 = 120 = 120 = 10

__________ _______ _____ ____

3! * (5 - 3)! 6 * 2! 6 * 2 12

5 things taken 3 at a time = 5 C 3 = 5!/[(5-3)!(3!)] = 5*4/2! = 20/2 = 10 combinations.

http://www.mathwords.com/c/combination_formula.htm

http://www.wolframalpha.com/input/?i=5+C+3

Note that 5! = 5*4*3!

so that the 3! in the denominator cancels out with the 3! in the numerator, and then you have

5!

------------ = 5*4/2! = 20/2 = 10

3! * (5 - 3)!

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Cx,y = x!

_________

y! * (x - y)!

I