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Permutation and Combination

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Table of Content:

Permutation and Combination Formulas
Definitions
Factorial
Permutation Formulas
Permutations of n Objects
Permutations of n Objects Taken r at a Time
Combination Formulas
Combinations of n Objects Taken r at a Time
Properties of Permutations and Combinations
Symmetry in Combinations
Repetition in Permutations
Special Cases
Permutation of Like Objects
Circular Permutations
Example Problems
Example 1: Finding Permutations
Example 2: Finding Combinations
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Permutation and Combination Formulas

Definitions

Permutation: Arrangement of objects in a specific order.

Combination: Selection of objects without considering the order.

Factorial

n! = n \times (n - 1) \times (n - 2) \times \dots \times 1

Permutation Formulas

Permutations of $n$ Objects

P (n) = n!

Permutations of $n$ Objects Taken $r$ at a Time

P (n, r) = \frac{n!}{(n - r)!}

Combination Formulas

Combinations of $n$ Objects Taken $r$ at a Time

C (n, r) = \frac{n!}{r! (n - r)!}

Properties of Permutations and Combinations

Symmetry in Combinations

C (n, r) = C (n, n - r)

Repetition in Permutations

P (n, r) with repetition = n^{r}

Special Cases

Permutation of Like Objects

If there are $n$ objects with $p_{1}$ of one kind, $p_{2}$ of another kind, ..., $p_{k}$ of $k$ kinds, the permutations are:

P (n; p_{1}, p_{2}, \dots, p_{k}) = \frac{n!}{p_{1}! \cdot p_{2}! \dots p_{k}!}

Circular Permutations

The number of ways to arrange $n$ objects in a circle:

P_{circular} (n) = (n - 1)!

Example Problems

Example 1: Finding Permutations

If there are 5 books and 3 are to be arranged on a shelf, how many arrangements are possible?

P (5, 3) = \frac{5!}{(5 - 3)!} = \frac{5!}{2!} = \frac{120}{2} = 60

Example 2: Finding Combinations

If there are 6 fruits and 2 are to be selected, how many selections are possible?

C (6, 2) = \frac{6!}{2! (6 - 2)!} = \frac{6!}{2! \cdot 4!} = \frac{720}{2 \cdot 24} = 15

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