In more detail
C(n, k) = n! / (k! (n − k)!), also written nCk or "n choose k". Choosing 3 of 10 people gives 120. Combinations are symmetric: C(n, k) = C(n, n − k), because picking 3 toppings out of 8 is the same as picking the 5 to leave out, and both give 56. When the same item may be chosen more than once, the count becomes C(n + k − 1, k): 3 scoops from 5 flavours with repeats allowed gives C(7, 3) = 35.
Why it matters
Combinations give lottery odds, poker-hand probabilities and the number of possible committees, samples or test cases. Using a Permutation formula when order does not matter overcounts by a factor of k!, which makes rare events look k! times more likely than they are.
Example
A lottery draws 6 numbers from 49:
C(49, 6) = (49 × 48 × 47 × 46 × 45 × 44) / 6!
= 10,068,347,520 / 720
= 13,983,816
One ticket therefore wins the jackpot with probability 1 in 13,983,816, about 7.15 × 10⁻⁸. The number of five-card poker hands from a 52-card deck is C(52, 5) = 2,598,960.
Common confusion
A Permutation counts ordered arrangements: P(10, 3) = 720 = 120 × 3!, because each group of 3 can be ordered in 3! = 6 ways.