Useful Relations in Permutations and Combination

1. Useful Relations - Factorial

n! = n.(n-1)!

2. n퐶푟= n푃푟/r! n n-1 3. Pr = n( Pr-1) 4. Useful Relations - Combinations n n 1. Cr = C(n - r)

Example

8 8 C6 = C2 = 8×72×1 = 28

n 2. Cn = 1 n 3. C0 = 1 n n n n n 4. C0 + C1 + C2 + ... + Cn = 2

Example

4 4 4 4 4 4  C0 + C1 + C2 + C3+ C4 = (1 + 4 + 6 + 4 + 1) = 16 = 2 n n (n+1)  Cr-1 + Cr = Cr (Pascal's Law)  n퐶푟 =n/퐶푟−1=n-r+1/r n n  If Cx = Cy then either x = y or (n-x) = y.

5. Selection from identical objects: Some Basic Facts

 The number of selections of r objects out of n identical objects is 1.  Total number of selections of zero or more objects from n identical objects is n+1.

6. Permutations of Objects when All Objects are Not Distinct

The number of ways in which n things can be arranged taking them all at a time, when st nd 푃1 of the things are exactly alike of 1 type, 푃2 of them are exactly alike of a 2 type, and th 푃푟of them are exactly alike of r type and the rest of all are distinct is

n!/ 푃1! 푃2! ... 푃푟!

1

Example: how many ways can you arrange the letters in the word THESE?

5!/2!=120/2=60

Example: how many ways can you arrange the letters in the word REFERENCE?

9!/2!.4!=362880/2*24=7560

7.Circular Permutations: Case 1: when clockwise and anticlockwise arrangements are different

Number of circular permutations (arrangements) of n different things is (n-1)!

1. Circular Permutations: Case 2: when clockwise and anticlockwise arrangements are not different

Number of circular permutations (arrangements) of n different things, when clockwise and anticlockwise arrangements are not different (i.e., when observations can be made from both sides), is 1/2(n−1)!

Sample Space and Probability 1. SETS Probability makes extensive use of set operations, so let us introduce at the Outset the relevant notation and terminology.

A set is a collection of objects, which are the elements of the set. If S is a set and x is an element of S, we write x ∈ S. If x is not an element of S, We write x ∉ S. A set can have no elements, in which case it is called the empty set, denoted by ɸ.

Sets can be specified in a variety of ways. If S contains a finite number of elements, say 푥1, 푥2, . . . , 푥푛, we write it as a list of the elements, in braces: S = { 푥1, 푥2, . . . , 푥푛}.

For example, the set of possible outcomes of a die roll is {1, 2, 3, 4, 5, 6}, and the set of possible outcomes of a coin toss is {H, T}, where H stands for “heads”

2

and T stands for “tails.”

If S contains infinitely many elements 푥1, 푥2, . . . ., which can be enumerated in a list (so that there are as many elements as there are positive integers) we write S = { 푥1, 푥2, . . . . .},  And we say that S is countably infinite. For example, the set of even integer can be written as {0, 2,−2, 4,−4, . . .}, and is countably infinity.  If every element of a set S is also an element of a set T, we say that S is a subset of T, and we write S ⊂ T or T ⊃ S. If S ⊂ T and T ⊂ S,  The two sets are equal, and we write S = T.  It is also expedient to introduce a universal set, denoted by Ω, which contains all objects that could conceivably be of interest in a particular context.

Set Operations

The complement of a set S, with respect to the universe Ω, is the set {x ∈ Ω |x ∉ S} of all elements of Ω that do not belong to S, and is denoted by 푆푐. Note that Ω 푐 = ɸ. The union of two sets S and T is the set of all elements that belong to Sor T (or both), and is denoted by S ∪ T. The intersection of two sets S and T is the set of all elements that belong to both S and T, and is denoted by S ∩ T. Thus, S ∪ T = {x | x ∈ S or x ∈ T}, S ∩ T = {x | x ∈ S and x ∈ T}. In some cases, we will have to consider the union or the intersection of several, even infinitely many sets, defined in the obvious way. For example, if for every positive integer n, we are given a set 푆푛 , then

∞ ⋃푛=1 푆푛 = 푆1 ∪ 푆2 ∪ … … .. = {x | x ∈ 푆푛 for some n}, and ∞ ⋂푛=1 푆푛 = 푆1 ∩ 푆2 ∩ … … .. = {x | x ∈ 푆푛 for all n}.

Two sets are said to be disjoint if their intersection is empty. More generally, Several sets are said to be disjoint if no two of them have a common element. A collection of sets is said to be a partition of a set S if the sets in the collection.

3

If x and y are two objects, we use (x, y) to denote the ordered pair of x and y. The set of scalars (real numbers) is denoted by Ɍ; the set of pairs (or triplets) of scalars, i.e., the two-dimensional plane (or three-dimensional space, respectively) is denoted by Ɍ2 (or Ɍ3, respectively).

Sets and the associated operations are easy to visualize in terms of Venn Diagrams

2. The Algebra of Sets Set operations have several properties, which are elementary consequences of the definitions. Some examples are: S ∪ T = T ∪ S , S ∪ (T ∪ U) = (S ∪ T) ∪ U, S ∩ (T ∪ U) = (S ∩ T) ∪ (S ∩ U) , S∪ (T ∩ U) = (S ∪ T) ∩ (S ∪ U), (S 푐)푐= S, S ∩ S 푐 = ɸ, S ∪Ω = Ω, S∩Ω = S. Two particularly useful properties are given by de Morgan’s laws which state that

푐 푐 (⋃푛 푆푛) = ⋂푛 푆푛 , 푐 푐 (⋂푛 푆푛) = ⋃푛 푆푛

PROOF: 푐 Suppose that x ∈ (⋃푛 푆푛) Then, x ∉ ⋃푛 푆푛 , which implies that for every n, we have x ∉ 푆푛. Thus, x belongs to the 푐 푐 푐 complement of every푆푛 , and 푥푛∈ ⋂푛 푆푛 . This shows that(⋃푛 푆푛) ⊂ ⋂푛 푆푛 . The converse inclusion is established by reversing the above argument, and the first law follows. The argument for the second law is similar.

4