The Combinations Calculator will find the number of possible combinations that can be obtained by taking a sub-set of items from a larger set. Basically, it shows how many different possible sub-sets can be made from the larger set. For this calculator, the order of the items chosen in the sub-set does not matter.
Calculate the combinations for C(n,r) = n! / ( r!(n - r)! ). For 0 <= r <= n.
“n choose r” = C(n,r) = n! / ( r! (n - r)! )
The formula shows the number of ways a sample of “r” elements can be obtained from a larger set of “n” distinguishable objects where order does not count and repetitions are not allowed.  "The number of ways of picking r unordered outcomes from n possibilities." 
Also referred to as r-combination or "n choose r". In some resources the notation uses k instead of r so you may see these referred to as k-combination or "n choose k".
Combination Problem 1: A teacher is going to choose 3 students from her class to compete in the spelling bee. She wants to figure out how many unique teams of 3 can be created from her class of 25.
In this example, we are taking a sub-set of 3 students (r) from a larger set of 25 students (n). Looking at the formula, we must calculate “25 choose 3”
C (25,3)= 25!/(3! * (25-3)!)= 2,300 Possible Teams
Combination Problem 2: A restaurant asks some of its frequent customers to choose their favorite 4 items on the menu. If the menu has 18 items to choose from, how many different answers could the customers give?
Here we take a 4 item sub-set (r) from the larger 18 item menu (n). Therefore, we must simply find “18 Choose 4”
C (18,4)= 18!/(4! * (18-4)!)= 3,060 Possible Answers
 Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 206, 2003.
For more information on combinations and binomial coefficients please see http://mathworld.wolfram.com/Combination.html