Fundamental Principles of Counting

8/11/2019 Fundamental Principles of Counting

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1. Fundamental Principles of Counting : Multiplication Theorem

If an operation can be performed in m different ways and following

which a second operation can be performed in n different ways, then thetwo operations in succession can be performed in m n different ways

2.

Fundamental Principles of Counting : Addition Theorem

If an operation can be performed in m different ways and a second

independent operation can be performed in n different ways, either ofthe two operations can be performed in (m+n) ways.

3.

Factorial

Let n be a positive integer. Then n factorial (n!) can be defined as

n! = n(n-1)(n-2)...1

Examples

i. 5! = 5 x 4 x 3 x 2 x 1 = 120

ii. 3! = 3 x 2 x 1 = 6

Special Cases

iii. 0! = 1

iv. 1! = 1

4.

Permutations

Permutations are the different arrangements of a given number ofthings by taking some or all at a time

Examples


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i. All permutations (or arrangements) formed with the letters a, b, c bytaking three at a time are (abc, acb, bac, bca, cab, cba)

ii. All permutations (or arrangements) formed with the letters a, b, c by

taking two at a time are (ab, ac, ba, bc, ca, cb)5. Combinations

Each of the different groups or selections formed by taking some or allof a number of objects is called a combination

Examples

i. Suppose we want to select two out of three girls P, Q, R. Then, possible

combinations are PQ, QR and RP. (Note that PQ and QP represent the

same selection)

ii. Suppose we want to select three out of three girls P, Q, R. Then, only

possible combination is PQR

6. Difference between Permutations and Combinations and How toAddress a Problem

Sometimes, it will be clearly stated in the problem itself whether

permutation or combination is to be used. However if it is not mentioned

in the problem, we have to find out whether the question is related to

permutation or combination.

Consider a situation where we need to find out the total number of

possible samples of two objects which can be taken from three objects

P,Q , R. To understand if the question is related to permutation orcombination, we need to find out if the order is important or not.

If order is important, PQ will be different from QP , PR will be differentfrom RP and QR will be different from RQ

If order is not important, PQ will be same as QP, PR will be same as RP

and QR will be same as RQ

Hence,

If the order is important, problem will be related to permutations.If the order is not important, problem will be related to combinations.


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For permutations, the problems can be like "What is the number of

permutations the can be made", "What is the number of arrangements

that can be made", "What are the different number of ways in whichsomething can be arranged", etc

For combinations, the problems can be like "What is the number ofcombinations the can be made", "What is the number of selections the

can be made", "What are the different number of ways in whichsomething can be selected", etc.

Mostly problems related to word formation, number formation etc will be

related to permutations. Similarly most problems related to selection of

persons, formation of geometrical figures , distribution of items (thereare exceptions for this) etc will be related to combinations.

7. Repetition

The term repetition is very important in permutations and combinations.

Consider the same situation described above where we need to find out

the total number of possible samples of two objects which can be takenfrom three objects P,Q , R.

If repetition is allowed, the same object can be taken more than once tomake a sample.

i.e., if repetition is allowed, PP, QQ, RR can also be considered aspossible samples.

If repetition is not allowed, then PP, QQ, RR cannot be considered aspossible samples

Normally repetition is not allowed unless mentioned specifically.

8.

pq and qp are two different permutations ,but they represent the same

combination.9.

Number of permutations of n distinct things taking r at a time

Number of permutations of n distinct things taking r at a time can be

given by

nPr= n!(nr)!=n(n1)(n2)...(nr+1)where 0rn


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If r > n, nPr= 0

Special Case:nP0= 1

nPr is also denoted by P(n,r).nPrhas importance outside combinatorics

as well where it is known as the falling factorial and denoted by (n) rornr

Examples

i.

8P2= 8 x 7 = 56

ii. 5P4= 5 x 4 x 3 x 2 = 120

10. Number of permutations of n distinct things taking all at atime

Number of permutations of n distinct things taking them all at a time= nPn= n!

11. Number of Combinations of n distinct things taking r at a time

Number of combinations of n distinct things taking r at a time ( nCr) can

be given by

nCr= n!(r!)(nr)!=n(n1)(n2)(nr+1)r!where 0rn

If r > n, nCr= 0

Special Case:nC0= 1

nCris also denoted by C(n,r).nCroccurs in many other mathematical

contexts as well where it is known as binomial coefficient and denotedby (nr)

Examples

i. 8C2= 8721= 28

ii. 5C4= 54324321= 5


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1. Useful Relations - Factorial

n! = n.(n-1)!

2. nCr=nPrr!

3.

Useful Relations - Permutations

1.

nPn= n!

2. nP0= 1

3.

nP1= n

4. nPn=nPn - 1

5. nPr= n(n-1Pr-1)

4. Useful Relations - Combinations1. nCr=

nC(n - r)

Example

i. 8C6=8C2= 8721= 28

2.

nCn= 1

3. nC0= 14.

nC0+nC1+

nC2+ ... +nCn= 2

n

Example

i. 4C0+4C1+

4C2+4C3+

4C4= (1 + 4 + 6 + 4 + 1) = 16 = 24

5. nCr-1+nCr=

(n+1)Cr (Pascal's Law)

6.

nCrnCr-1=n-r+1r7. If nCx=

nCy then either x = y or (n-x) = y

5. Selection from identical objects: Some Basic Facts

i. The number of selections of r objects out of n identical objects is 1

ii. Total number of selections of zero or more objects from n identical

objects is n+1.


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Fundamental Principles of Counting

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