A combination is a selection of elements from a set where the order of selection does not matter.
Order doesn't matter means that the selections AB and BA are considered a single combination (a single selection).
If the order does matter such as in a digital lock (pin) or the arrival order of a race, the term used is permutation
The most known example is a lottery - if the number are selected in the bad order, you still win.
When repeat is not valid (ie AA is not a valid pair)
We say that:
The best known example of a combination without repetition is lottery numbers (2,14,15,27,30,33)
Combination calculation without repetition is also known as:
Without repetition, the number of combination possible of length k in a set of possible value of length n is: <MATH> \binom nk = (n \text{ choose } k) = \frac{n(n-1)\dotsb(n-k+1)}{k(k-1)\dotsb1} = \frac{n!}{k!(n-k)!} </MATH>
Note:
n! is factorial n
Therefore, when the length of the set is equal to the length of the combinations, the number of combinations is 1.
Combination where repetition is allowed is also known as:
Example: coins in your pocket (5,5,5,10,10)
There are <math>\tbinom {n+k-1}k</math> ways to choose k elements from a set of n elements if repetitions are allowed.
<MATH> \binom {n+k-1}k = ({n+k-1} \text{ choose } k) = \frac{(k+n-1)!}{k!(n-1)!} </MATH>