◀ ▲ ▶Branches / Combinatorics / Part: Cycles, Permutations, Combinations and Variations

Part: Cycles, Permutations, Combinations and Variations

The most common combinatorial operations are cycles, permutations, combinations and variations. Loosely speaking, and before we define these three operations more rigidly, cycles are circular arrangements of objects, in which there is no specific "first" or "last" object, but the order of the objects inside the cycle is important. Cycles are essentially the same as permutations with the difference, that the latter are arrangements of objects with distinguished first and last elements. 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 Cycles

• Ways for $n$ persons to sit around a round table.

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

Table of Contents

1. Definition: Cycles
2. Definition: Permutations
3. Theorem: Approximation of Factorials Using the Stirling Formula
4. Definition: Combinations
5. Proposition: Factorial

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