Permutations | Moosa Academy

A permutation is an arrangement where order matters. Unlike a combination, ABC and BAC count as two different permutations — from the three letters there is one combination but six permutations.

Permutations Arrangements, in order

A permutation chooses elements from with regard to order. The same elements in a new order make a new permutation.

From , the arrangements of all three are ABC, ACB, BAC, BCA, CAB, CBA — six permutations, but only one combination .

Formula The three laws

Without repetition (each element used once):

With repetition (an element may be reused):

Arranging all

(when

Allowing repetition always gives a larger count, since no choice is ever used up.

Example Filling the slots

A 4-digit job number from the digits 1–9, no digit repeated. Each slot has one fewer choice than the last:

⟹ 3,024 numbers.

Example A few more

Arranging 3 books chosen from 7

⟹ 210 arrangements.

A 3-letter password from {A, B, C, D, E}, no repeats

⟹ 60 passwords.

A 5-digit security code from 0–9, repetition allowed

Every position has all 10 digits:

⟹ 100,000 codes.

Arranging all 5 letters of a word (all different)

⟹ 120 arrangements.

Reference The three types

Type	Formula	Example	Result
No repetition	n! ÷ (n − r)!	P(9, 4)	3,024
With repetition	nʳ	9⁴	6,561
Complete (r = n)	n!	P(5, 5)	120

Summary

Permutations arrange of where order matters: ABC ≠ BAC.
Without repetition: — each element is used once.
With repetition: — always a larger count.
Arranging all elements (when ) gives .