UNIT-1

Permutation and Combination

Points to be covered:

• Meaning,

• Fundamental principle of counting,

• Theorem based on permutation and combination (without proof),

• Permutations of things not all different, Permutation when repetition is allowed, circular permutations,

• examples of permutation and combination.11

Fundamental Principle of Counting

The rules of sum and product:

(1) The Rule of Sum:

• If a first task can be performed in m ways,

while a second task can be performed in n

ways, and the tasks cannot be performed

simultaneously, then performing either task

can be accomplished in any one of m+n ways.

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(2) The Rule of Product

• If a procedure can be broken down into first

and second stages, and if there are m possible

outcomes for the first stage and if, for each of

these outcomes, there are n possible outcomes

for the second stage, then the total procedure

can be carried out, in the designated order, in

mn ways.

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Factorial:

• The factorial function (symbol: !) means to

multiply a series of descending natural

numbers.

Examples:

(1) 4! = 4 × 3 × 2 × 1 = 24

(2) 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

(3) 1! = 1

1 4

n n!

1 1 1 1

2 2 × 1 = 2 × 1! = 2

3 3 × 2 × 1 = 3 × 2! = 6

4 4 × 3 × 2 × 1 = 4 × 3! = 24

5 5 × 4 × 3 × 2 × 1 = 5 × 4! = 120

6 etc etc

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Permutation:

• The different arrangements of a given number

of things by taking some or all at a time, are

called permutations.

• In Permutation, ordered is an important.

• A formula for the number of possible

permutations of r objects from a set of n.

• This is usually written nPr .

6

!

( )!n r

nP

n r

For example:

I have three letters A,B and C. and I want to

make a word of three letters. Then possible

permutations (arrangements) are:

.

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Important Results:

8

0

( 1) ( 1) ( 1)

1 1

1. P !

2. 1

3. .

4. .

n n

n

n r n r n r

n r n r n r

n

P

P P r P

P r P P

Example:

Find how many four-letter words can be formed

out of the letters of the word ‘ ARIEL’

Answer:

There are 5 different letters in the word ‘ARIEL’.

The total number of four letter words, which can

be formed out of these 5 letters

= 5P4 = 5 * 4* 3* 2 = 3024.

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TYPES OF PERMUTATION:

(1)Repetition is allowed

(2)Repetition is not allowed

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(1) Repetition is allowed:

(when all things are different)

Example: How many 3 letter words can be formed using the letters c,a,t allowing for repetition of the letters?

Solution:For this problem, 3 locations are needed:

_____ • _____ • _____There are 3 letters which can be used to fill the first location. Because repetition is allowed, the same 3 letters can be used to fill the second location and also the last location.

__3___ • __3___ • __3___ = 27 arrangements

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27 arrangements with repetition:

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ccc cat ctc aaa act ata ttt tac tat

caa cta cca acc atc aac taa tca ttc

ctt cac cct att aca aat tcc tct tta

Permutations of things not all different

• The number of permutations of ‘n’ things of which ‘p’

things are alike and of one kind, ‘q’ things are alike

and of another kind, ‘r’ things are alike and of another

kind is given by

• If out of ‘n’ things, ‘r’ things are alike and of one kind

and (n-r) things are alike and of another kind, then the

total number of permutations

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!

! ! !

n

p q r

!

!( )!

n

r n r

• Example:

1. How many different 5-letter words can be

formed from the word APPLE ?

Answer:

You divide by 2! because the letter P repeats

twice

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5 5 5*4*3*2*160

2! 2

P

Example:

How many different words can be made out of the

word ‘ ALLAHABAD’?

Answer:

The word ‘ ALLAHABAD’ consists of 9 letters in

which A is repeated four times, L is repeated

twice and the rest are all different.

Hence the required number of word is

9! / 4!2! = (9*8*7*6*5*4!)/ 4!*2

= 7560.

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(2) Repetition is not allowed

Example :

• Consider a lottery in which 6 balls are consecutively drawn at

random from an urn containing 99 balls, each printed with a

unique number 1, ..., 99. What are the total number of possible

outcomes of this draw?

Answer:

Each drawing is a permutation of 6 numbers chosen from a set of 99,

without repetition. Thus, the total number of possible permutations

is:

99P6 = 99 × 98 × 97 × 96 × 95 × 94

= 806, 781, 064, 320.

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Circular Permutation

• The circular permutations are used when the

elements have to be arranged "in a circle"

order, (for example, the guests around a table

at a dinner party), so that the first element that

"is located" in the sample determines the

beginning and the end of the sample.

PCn = (n-1)!

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How can A, B, C be arranged around a circle?

Not in the three ways as shown above! Why?

Because each one of A, B, C has the same neighbor!

Without changing neighbor, only changing seats will not change the circular permutation.

Change neighbors and you will change the circular permutation. As follows:

So, three persons A, B, C can only be arranged in 2 ways around a circle.

In a formula type it is:

(3-1)! = 2! = 2 ways

Combination:

• The different groups or collections (or

selections) that can be formed out of a given set

of things by taking some or all of them at a time

(without consider to the order of their

arrangement) are called their combinations.

• In combinations, order does not matter.

• It is denoted by

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(n, r)n r

nor C orC

r

Where n= total numbers

r = no. of wish

20

!

!(n r)!

n n

r r

Important Result:

21

0

(n 1) ( 1)

1 1

1. 1

2. 1

3.

4.r . C .[ ]

5. C

n

n n

n r n n r

n r r

n r n r n r

C

C

C C

n C

C C

EXAMPLE:

How many ways can 3 men select out of 5 men.

Answer:

Here n=5 and r = 3.

By using formula

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! 5! 5*4*3!10

!( )! 3!(5 3)! 3!*2!

n

r n r

Difference between Permutation and

Combination

Sr.

No.

Permutation Combination

1. Use for arrangement Use for selection

2. Order is important Order is not an important

3. nPr = n!/ (n-r)! nCr = n! / r! (n-r)!

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References:

• Sile-2,3: www.mgt.ncu.edu.tw/~ylchen/dismath/chap01.pp

• Slide-4,5: www.mathsisfun.com/numbers/factorial.html

• Slide-6: http://www.indiabix.com/aptitude/permutation-and-combination/formulas

• IMAGE-1-SLIDE-7: https://www.google.co.in/search?hl=en&site=imghp&tbm=isch&source=hp&biw=1366&bih=643&q=PERMUTATION&oq=PERMUTATION&gs_l=img.3..0l10.2088.5909.0.6413.11.11.0.0.0.0.492.671.0j1j4-1.2.0.msedr...0...1ac.1.60.img..9.2.670.3cYXg9YGhvo#facrc=_&imgdii=_&imgrc=UTWyBH6hERKNgM%253A%3BkA_jTOu4e4nicM%3Bhttp%253A%252F%252Fwww.mathwarehouse.com%252Fprobability%252Fimages%252Fmultiplication-principle4.gif%3Bhttp%253A%252F%252Fwww.mathwarehouse.com%252Fprobability%252Fpermutations-repeated-items.php%3B448%3B201

• Slide 11-12: www.regentsprep.org/regents/math/algebra/apr2/LpermRep.htm

• Slide-17: http://www.vitutor.com/statistics/combinatorics/circular_permutations.html

• Slide-18: http://www.math-for-all-grades.com/CircularPermutation.html

• Business Mathematics by G.C. Patel and A.G. Patel by atul prakashan

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