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Permutation

See how to solve a permutation nPr
(+ permutation with repetition, circular permutation, bracelet permutation).
8 examples and their solutions.

Permutation

Formula

r numbers nPr = n⋅(n - 1)⋅(n - 2)⋅ ...
nPr: Starting from n, multiply r numbers.
Meaning: From n things,
pick & arrange r things.

Combination (Math)
nPn = n! - [1]
nP1 = n - [2]
nP0 = 1 - [3]
[1]
Starting from n, multiply n numbers.
→ n!
Factorial
[2]
Starting from n, multiply 1 number.
→ n
[3]
From n things, pick & arrange 0 things.
→ Don't do anything.
→ 1 way

Example

6P4
Solution

Example

9 students
Find the number of ways to choose 3 students and arrange them in a row.
Solution

Example

Numbers: 1, 2, 3, 4, 5, 6, 7, 8
By using the numbers once,
find the number of ways to make a 4-digit number.
Solution

Example

Numbers: 0, 1, 2, 3, 4, 5
By using the numbers once,
find the number of ways to make a 3-digit number.
Solution

Permutation with Repetition

Formula

N = n!p!q!r!⋅...
n = p + q + r + ...
p, q, r: Numbers of identical things

Example

Letters: a, a, a, b, b, c, c
By using each letter once,
find the number of ways to make a 7-letter word.
Solution

Example

Find the number of shortest paths to move from A to B.

Solution

Circular Permutation

Definition

Circular permutation is used
when arranging things in a circle.
These two cases are the same case
because when you rotate the left case ↷,
you can get the right case.

Formula

N = (n - 1)!

Example

5 students
Find the number of ways to make them sit at a round table.
Solution

Bracelet Permutation

Definition

Bracelet permutation is used
when arranging things in a bracelet (or a necklace).
These two cases are the same case
because when you flip the left case,
you can get the right case.

Formula

N = (n - 1)!2

Example

7 beads
Find the number of ways to make a bracelet by using these beads.
Solution