Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages

Basic Quantitative Techniques

ABS-Bangalore

1Basic Quantitative Techniques - RVMReddy - ABSApril 8, 2023

Dr. R. Venkatamuni ReddyAssociate Professor

Contact: 09632326277, 080-30938181

[email protected]

[email protected]


Permutations and

Combinations

April 8, 2023 3Basic Quantitative Techniques - RVMReddy - ABS

PermutationsPermutations refers to the different ways in which a

number of a number of objects can be arranged in a different order

Example: Suppose there are two things x and y, they can be arranged in to two different ways i.e,. xy and yx . These two arrangements is called permutation

Similarly x, y and z

xyz, xzy, yxz, yzx, zxy, zyx is 6 arrange permutation

(if we want to have two things only from x,y,z then xy,xz,yz,yx,zx,yz only in this case)April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 4

“The word permutation thus refers to the arrangements which can be made by taking some or all of a number of things”

Formulae 1: Finding the number of permutations of ‘n’ dissimilar things taken ‘r’ at a time

n=number of different things given, r=number of different things taken at a time out of different things givenApril 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 5

Permutations

)1)...(2)(1(),( rnnnnrnP

)!(

!

rn

n

Example 1: There are six boxes and three balls. In how many ways can these three balls be discretely put into these six boxes.

Solution:

Example 2: How many four-letter words can be made using the letters of the word ‘BANGALORE’ and ‘ALLIANCE’

Solution: n=9, r=4 and n=8, r=4 April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 6

Permutations

120)!36(

!6),(

rnP

Example 3: How many arrangements are possible of the letters of the words ‘JAIPUR’, ‘BANGALORE’ and ‘ALLIANCE’

Hint: n=6, r=6 and n=9, r=9 and n=8,r=8

April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 7

Permutations

Formulae 2: Finding the number of permutations of ‘n’ things taken ‘r’ at a time, given that each of the elements can be repeated once, twice….up to ‘r’ times

Or

‘n’ things taken all at a time of which ‘p’ are alike, ‘q’ others are alike and ‘r’ others alike

Example 1: How many permutations are possible of the letters of the word PROBABILITY when taken all at a time? April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 8

Permutations

!!.!.

!),(

rqp

nrnP

Solution: n=11, p=2 ( as letter B is occurring twice in the given word) , and q=2 ( as letter I is occurring twice in the given word)

And all other letters in the given word are different. The required number of permutations is (r is not valid in this)

Example 2: You are given a word “MANAGEMENT” and asked to compute the number of permutations that you can form taking all the letters from this word?


Permutations

!!.!.

!),(

rqp

nrnP 9979200

!2!.2

!11

Permutations

10Basic Quantitative Techniques - RVMReddy - ABS

Permutation formula proof There are n ways to choose the first element

– n-1 ways to choose the second– n-2 ways to choose the third– …– n-r+1 ways to choose the rth element

By the product rule, that gives us:

P(n,r) = n(n-1)(n-2)…(n-r+1)


Combinations refers to the number of arrangements which can be made from a group of things irrespective of the order

Combinations differ from permutations in that one combination such as xyz may be stated in the form of several permutations just by rearranging the orders as : xyz, xzy, yxz, yzx, zxy, zyx

Note: All of these are one combination but they are six permutations

IMP Note: The number of permutations is always greater than the number of combinations in any given situation since a combination of n different things can be generate n factorial permutations


Combinations

Formulae 1: The number of r-combinations of a set with n elements, where n is non-negative and 0≤r≤n is:

n= number of different things given

r= number of different things taken at a time out of different things given


)!(!

!),(

rnr

nrnC

Combinations

Example 1: in how many ways can four persons be chosen out of seven?

n=7, r=4

Example 2: Find the number of combinations of 50 things taking 46 at a time. ANS: 230300


Combinations

35)!47(!4

!7

)!(!

!),(

rnr

nrnC

Formulae 2: The number of ways in which x+y+z things can be divided into three groups contain x, y, and z things respectively is

Example: In how many ways can 10 books be put to three shelves which can contain 2, 3 and 5 books respectively?


Combinations

!!.!.

)!(),(

zyx

zyxrnC

2520!5!.3!.2

)!532(),(

rnC

Combinations


Combinations

How many different poker hands are there (5 cards)?

How many different (initial) blackjack hands are there?

2,598,960!47*1*2*3*4*5

!47*48*49*50*51*52

!47!5

!52

)!552(!5

!52)5,52(

C

1,3261*2

51*52

!50!2

!52

)!252(!2

!52)2,52(

C


Combination formula proof Let C(52,5) be the number of ways to generate

unordered poker hands The number of ordered poker hands is P(52,5) =

311,875,200 The number of ways to order a single poker hand

is P(5,5) = 5! = 120 The total number of unordered poker hands is the

total number of ordered hands divided by the number of ways to order each hand

Thus, C(52,5) = P(52,5)/P(5,5)


Combination formula proof Let C(n,r) be the number of ways to generate

unordered combinations The number of ordered combinations (i.e. r-

permutations) is P(n,r) The number of ways to order a single one of those

r-permutations P(r,r) The total number of unordered combinations is the

total number of ordered combinations (i.e. r-permutations) divided by the number of ways to order each combination

Thus, C(n,r) = P(n,r)/P(r,r)


Combination formula proof

Note that the textbook explains it slightly differently, but it is same proof

)!(!

!

)!/(!

)!/(!

),(

),(),(

rnr

n

rrr

rnn

rrP

rnPrnC


Let n and r be non-negative integers with r ≤ n. Then C(n,r) = C(n,n-r)

Proof:

)!(!

!),(

rnr

nrnC

! )()!(

!),(

rnnrn

nrnnC

)!(!

!

rnr

n


Combination formula proof

Binomial Coefficients

The expression x +y is a binomial expression as it is the sum of two terms.

The expression (x +y)n is called a binomial expression of order n.



Pascal’s Triangle – The number C(n, r) can be obtained by constructing a

triangular array.– The row 0, i.e., the first row of the triangle, contains

the single entry 1. The row 1, i.e., the second row, contains a pair of entries each equal to 1.

– Calculate the nth row of the triangle from the preceding row by the following rules:


Binomial Coefficients The technique known as divide and conquer can be

used to compute C(n, r ). In the divide-and-conquer technique, a problem is

divided into a fixed number, say k, of smaller problems of the same kind.

Typically, k = 2. Each of the smaller problems is then divided into k smaller problems of the same kind, and so on, until the smaller problem is reduced to a case in which the solution is easily obtained.

The solutions of the smaller problems are then put together to obtain the solution of the original problem.April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 29

Thank You


Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages

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