Arrangement:
n items can be arranged in n! ways.
Arrangement with repetitions:
If x items out of n items are repeated, then the number of ways of arranging these n items is 
ways.
If a items, b items, and c items are repeated within n items, then they can be arranged in 
ways.
Circular arrangement:
The number of ways of arranging n items around a circle is 1 for n = 1, 2, and (n − 1)! for n ≥ 3.
If it’s a necklace or bracelet that can be flipped over, then the possibilities are (n − 1)!/2
Permutation:
It is a way of selecting and arranging r objects out of a set of n objects.
Combination:
It is a way of selecting r objects out of n (arrangement does not matter).
- Selecting r objects out of n is the same as selecting (n − r) objects out of n,
- The total selections that can be made from ‘n’ distinct items is given by
Partitioning:
- The number of ways to partition n identical things in r distinct slots is given by
- The number of ways to partition n identical things in r distinct slots so that each slot gets at least 1 is given by
- The number of ways to partition n distinct things in r distinct slots is given by rn
- Number of ways to partition n distinct things in r distinct slots where arrangement matters is given by
Integral solutions:
- The number of positive integral solutions to x1 + x2 + x3 +...+ xn = k where k ≥ 0 is
- The number of non-negative integral solutions to x1 + x2 + x3+...+ xn = k where k ≥ 0 is
Derangements:
If n distinct items are arranged, then the number of ways in which they can be arranged so that they do not occupy their intended spot is
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