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Permutation Types and Examples

There are two types of permutations: 1) with repetition allowed, where the same object can be selected multiple times, and 2) without repetition, where each object can only be selected once. Permutations with repetition are calculated using factorial notation (n!), while permutations without repetition use the formula nPr = n!/(n-r)!. The document provides examples of calculating permutations in different scenarios, including with identical objects.

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Riven Guadalquiver Barquilla Lpt
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133 views1 page

Permutation Types and Examples

There are two types of permutations: 1) with repetition allowed, where the same object can be selected multiple times, and 2) without repetition, where each object can only be selected once. Permutations with repetition are calculated using factorial notation (n!), while permutations without repetition use the formula nPr = n!/(n-r)!. The document provides examples of calculating permutations in different scenarios, including with identical objects.

Uploaded by

Riven Guadalquiver Barquilla Lpt
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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Permutation

There are basically two types of permutation:


1. Repetition is Allowed: such as the lock above. It could be "333".
2. No Repetition: for example the first three people in a running race. You can't be first and second.
But we need to consider also the FACTORIAL NOTATION, PERMUTATION WITH IDENTICAL OBJECTS, and
CIRCULAR PERMUTATION

1. Permutations with Repetition


The number of n permutations taken n at a time is denoted nPn
The total number of permutations of a set of n object is given by:

𝑛𝑃𝑛 = 𝑛(𝑛 − 1) ∙ (𝑛 − 2) ∙. . .∙ 3 ∙ 2 ∙ 1

Example:
1. 2𝑃2 = 2 ∙ 1 or AB = AB, BA
2. 3𝑃3 = 3 ∙ 2 ∙ 1 or ABC = ABC, BAC, CBA, BCA, CAB, ACB
3. 4𝑃4 = 43 ∙ 2 ∙ 1 or ABCD =
4. In how many ways can six books be arranged on a shelf?
5. 12𝑃12 = 12 ∙ 11 ∙ 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
= 12 !
FACTORIAL NOTATION
The product 4 ∙ 3 ∙ 2 ∙ 1 can be written simply as 4 ! (Read as four factorial)

Example:
1. 12𝑃12 = 12 ∙ 11 ∙ 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
= 12 !
= 479,001,600

2. Permutations without Repetition


Permutation of n different objects taken r at a time is given by
𝑛!
𝑛𝑃𝑟 = (𝑛−𝑟)!

Example:
3!
1. 3𝑃1 = (3−1)! or ABC = A, B, C
=3
4!
2. 4𝑃2 = (4−2)!
or ABCD = AB, BA, CA, DA, AC, BC, CB,
DB, AD, BD,
= 12 CD, DC

4!
3. 4𝑃3 = (4−3)!
or ABCD = ABC, BCA, CBA, BAC, CAB,
ACB
= 24 BCD, CBD, DBC, BDC, CDB,
DCB
CDA, DAC,
CAD, DCA, ADC, ACD
DAB, BAD,
BDA, DBA, ABD, ADB
5!
4. 5𝑃2 = (5−2)! or ABCDE =
= 20
8!
5. 8𝑃2 = (8−2)!
or 12345678 =
= 56

PERMUTATION WITH IDENTICAL OBJECTS

Example:
Roxanne has three vases of the same kind and two candle stands of the same kind. In how many ways can she arrange
these items in a line?

Sol.

5! 5∙4∙3! 20
= = = 10
3!2! 3! 2∙1 2

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