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Permutations, Combinations and Variations

The most common combinatorial operations are permutations, combinations and variations. Loosely speaking, and before we define these three operations more rigidly, permutations are (finite or infinite) arrangements of objects. Here, the order of the objects is important. Combinations are the different ways to pick a finite number of objects out of another finite number of objects. Unlike for permutations, in combinations, the order of objects picked usually does not play any role. A combination, in which order does play a role, is called a variation.

Examples of Permutations

Ways to put $n$ books into a specific order on a shelf.

Different strings we can build using the $26$ letters of the Latin alphabet $a,b,\ldots, z,$ such that each letter is used only once in a single string.

Orders, in which $n$ people can enter a door.

Examples of Combinations

Ways, in which $3$ out of $10$ sportspeople can win a medal in a competition (no matter whether gold, silver, or bronze).

Possibilities to choose $2$ representatives out of $100$ students.

Different outcommings rolling $3$ identical dice.

Examples of Variations

Ways, in which $3$ out of $10$ sportspeople can win a medal in a competition, the first winning gold, the next silver, and the third bronze.

Possibilities to choose $2$ representatives out of $100$ students, one as the “president” and the other as the “vice-president”.

Different outcommings rolling $3$ dice which are distinguishable by color, e.g. white, red, and black dice.