Aptitude Permutation And Combination Tutorials & Tricks

Permutation And Combination Tutorials & Tricks

Introduction

Permutation:An arrangement is called a Permutation. It is the rearrangement of objects or symbols into distinguishable sequences. When we set things in order, we say we have made an arrangement. When we change the order, we say we have changed the arrangement. So each of the arrangement that can be made by taking some or all of a number of things is known as Permutation.

Combination:A Combination is a selection of some or all of a number of different objects. It is an un-ordered collection of unique sizes.In a permutation the order of occurence of the objects or the arrangement is important but in combination the order of occurence of the objects is not important.

Formula for finding out Permutation And Combination

1. Factorial Notation: Let n be a positive integer. Then, factorial n, denoted n! is defined as: n! = n(n - 1)(n - 2) ... 3.2.1.

2. Permutations: The different arrangements of a given number of things by taking some or all at a time, are called permutations

3. Number of Permutations Number of all permutations of n things, taken r at a time, is given by: nPr = n(n - 1)(n - 2) ... (n - r + 1) = n!/(n - r)!

4. An Important Result: If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind;p3 are alike of third kind and so on and pr are alike of rth kind, such that (p1 + p2 + ... pr) = n. Then, number of permutations of these n objects is = n!/(p1!).(p2)!.....(pr!)

5. Combinations: Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination. Then, number of permutations of these n objects is = n!/(p1!).(p2)!.....(pr!)

Examples: 1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA. Note: AB and BA represent the same selection.

2. All the combinations formed by a, b, c taking ab, bc, ca.

3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.

4. Various groups of 2 out of four persons A, B, C, D are: AB, AC, AD, BC, BD, CD.

5. Note that ab ba are two different permutations but they represent the same combination. Number of Combinations: The number of all combinations of n things, taken r at a time is: nCr = n!/(r!)(n - r)! = n(n - 1)(n - 2) ... to r factors / r! .

6. 1) Permutation = nPr = n! / (n-r)!

2) Combination = nCr = nPr / r! where, n, r are non negative integers and r<=n. r is the size of each permutation. n is the size of the set from which elements are permuted. !is the factorial operator.

Sample Example

Ex

In how many ways can the letters of the word PATANA can be arranged

A

 PATANA has 6 letters 1P, 3A, 1T, 1N No of arrangement = 6!/(1! 3! 1! 1!) = 120

Ex

How many words can be formed from the letters of the word " ENGINEERING" , so that vowels always come together ?

A

 Word ENGINEERING has 11 letters, from which EIEEI are vowels, they can e treated as single letter (EIEEI)NGNRNG, So seven letters has 3 N, 2 G , 1 R and single (EIEEI) No of arrangements = 7!/(2! 3!) = 420 arrangement of EIEEI = 5!/(2! 3!) = 10 total number of arrangemnets = 420 * 10 = 4200 (by rule of multiplication) 

Ex

In how many different ways can the letters of the word COMPUTER can be arranged in such a way that vowels may occupy only odd positions ?

A

 Here odd and even positions are : C O M P U T E R (O) (E) (O) (E) (O) (E) (O) (E) Now 3 vowels O, U , E In 5 odd place 3 vowels arranged as 5P3 = 5!/2! =60 Also remaning 5 places can be arranged by C, M, P, T, R Remaning 5 consonants arranged as 5P5 = 5! =120 ways So, required number of ways= 120 * 60 = 7200 

Ex

In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women ?

A

 Required number of ways = 7C5 * 3C2 = 7!/(5! 2!) * 3!/(2! 1!) = 63