# GMAT 2013 Exam, Preparation & Important Dates

24504 Followers - 240 Articles - 865 Questions and Answers

# Permutations and Combinations Basics

by Suresh

 FUNDAMENTAL PRINCIPLE OF COUNTING If an operation can be performed in 'm' different ways and another operation in 'n' different ways then these two operations can be performed one after the other in 'mn' waysIf an operation can be performed in 'm' different ways and another operation in 'n' different ways then either ofthese two operations can be performed in 'm+n' ways.(provided only one has to be done)

This principle can be extended to any number of operations

FACTORIAL 'n'

The continuous product of the first 'n' natural numbers is called factorial n and is deonoted by n! i.e, n! = 1×2×3x ….. x(n-1)xn.

PERMUTATION

An arrangementthat can be formed by taking some or all of a finite set of things (or objects) is called a Permutation.Order of the things is very important in case of permutation.A permutation is said to be a Linear Permutation if the objects are arranged in a line. A linear permutation is simply called as a permutation.A permutation is said to be a Circular Permutation if the objects are arranged in the form of a circle.The number of (linear) permutations that can be formed by taking r things at a time from a set of n distinct things is denoted by .%

NUMBER OF PERMUTATIONS UNDER CERTAIN CONDITIONS

1. Number of permutations of n different things, taken r at a time, when a particular thng is to be always included in each arrangement , is .

2. Number of permutations of n different things, taken r at a time, when a particular thing is never taken in each arrangement is .

3. Number of permutations of n different things, taken all at a time, when m specified things always come together is .

4. Number of permutations of n different things, taken all at a time, when m specified never come together is .

5. The number of permutations of n dissimilar things taken r at a time when k(< r) particular things always occur is .

6. The number of permutations of n dissimilar things taken r at a time when k particular things never occur is .

7. The number of permutations of n dissimilar things taken r at a time when repetition of things is allowed any number of times is

8. The number of permutations of n different things, taken not more than r at a time, when each thing may occur any number of times is .

9. The number of permutations of n different things taken not more than r at a time .

 + PERMUTATIONS OF SIMILAR THINGS+ The number of permutations of n things taken all tat a time when p of them are all alike and the rest are all different is .If p things are alike of one type, q things are alike of other type, r things are alike of another type, then the number of permutations with p+q+r things is .

CIRCULAR PERMUTATIONS

}1. The number of circular permutations of n dissimilar things taken r at a time is .

2. The number of circular permutations of n dissimilar things taken all at a time is .

3. The number of circular permutations of n things taken r at a time in one direction is .

4. The number of circular permutations of n dissimilar things in clock-wise direction = Number of permutations in anticlock-wise direction = .

COMBINATION

A selection that can be formed by taking some or all of a finite set of things( or objects) is called a Combination

The number of combinations of n dissimilar things taken r at a time is denoted by .

1.

2.

3.

4.

5. The number of combinations of n things taken r at a time in which

a)s particular things will always occur is .

b)s particular things will never occur is .

c)s particular things always occurs and p particular things never occur is .

DISTRIBUTION OF THINGS INTO GROUPS

1.Number of ways in which (m+n) items can be divided into two unequal groups containing m and n items is .

2.The number of ways in which mn different items can be divided equally into m groups, each containing n objects and the order of the groups is not important is

3.The number of ways in which mn different items can be divided equally into m groups, each containing n objects and the order of the groups is important is .

4.The number of ways in which (m+n+p) things can be divided into three different groups of m,n, an p things respectively is

5.The required number of ways of dividing 3n things into three groups of n each =.When the order of groups has importance then the required number of ways=

DIVISION OF IDENTICAL OBJECTS INTO GROUPS

The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is

}The number of non-negative integral solutions of the equation .

The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is

The number of positive integral solutions of the equation .

The number of ways of choosing r objects from p objects of one kind, q objects of second kind, and so on is the coefficient of in the expansion

he number of ways of choosing r objects from p objects of one kind, q objects of second kind, and so on, such that one object of each kind may be included is the coefficient of is the coefficient of in the expansion
.

 %{font-family:verdana}+*TOTAL NUMBER OF COMBINATIONS*+% %{font-family:verdana}1.The total number of combinations of things taken any number at a time when things are alike of one kind, things are alike of second kind….things are alike of kind, is .% %{font-family:verdana}2.The total number of combinations of things taken one or more at a time when things are alike of one kind, things are alike of second kind….things are alike of kind, is% .

SUM OF THE NUMBERS

Sum of the numbers formed by taking all the given n digits (excluding 0) is

Sum of the numbers formed by taking all the given n digits (including 0) is

Sum of all the r-digit numbers formed by taking the given n digits(excluding 0) is %

%{font-family:verdana}Sum of all the r-digit numbers formed by taking the given n digits(including 0) is

DE-ARRANGEMENT:

The number of ways in which exactly r letters can be placed in wrongly addressed envelopes when n letters are placed in n addressed envelopes is .

The number of ways in which n different letters can be placed in their n addressed envelopes so that al the letters are in the wrong envelopes is .

IMPORTANT RESULTS TO REMEBER

In a plane if there are n points of which no three are collinear, then

1. The number of straight lines that can be formed by joining them is .

2. The number of triangles that can be formed by joining them is .

3. The number of polygons with k sides that can be formed by joining them is .

In a plane if there are n points out of which m points are collinear, then

1. The number of straight lines that can be formed by joining them is .

2. The number of triangles that can be formed by joining them is .

3. The number of polygons with k sides that can be formed by joining them is .

Number of rectangles of any size in a square of n x n is

In a rectangle of p x q (p < q) number of rectangles of any size is

In a rectangle of p x q (p < q) number of squares of any size is

n straight lines are drawn in the plane such that no two lines are parallel and no three lines three lines are concurrent. Then the number of parts into which these lines divide the plane is equal to .

Image Credits: cristic, cosmolallie, farouqtaj, churl

Vote
Current Rating
1
Rate Up
Rate Down
rajesh2953Tue, 14 May 2013 03:27:52 -0000

The number of circles can be drawn out of 10 points of which 7 are collinear will be:

Vote
Current Rating
1
Rate Up
Rate Down
MahmuddgThu, 18 Apr 2013 10:10:34 -0000

Combination a selection that can be formed by taking some or all a finite set of thing (or object) is called a combination

Vote
Current Rating
0
Rate Up
Rate Down
obyejekwuMon, 04 Mar 2013 20:40:47 -0000

Hi! Pls help me with this question. Six papers are set in an examination of which two are mathematical. In how many orders can the papaers be arranged so that: I) the two mathematical papers are together ii) the two mathematical papers are not consecutive

Rating
1
Rate Up
TALENTED BOYTue, 19 Mar 2013 02:36:26 -0000

The answer is (i) (n-1)! (ii) n!

Rating
1
Rate Up
devdarguyThu, 21 Mar 2013 16:24:33 -0000

how the second one is n!.

Vote
Current Rating
1
Rate Up
Rate Down
jbenTue, 19 Feb 2013 07:59:40 -0000

In how many different ways can 6 men and 4 women be seated in a straight line so that not two women are seated together?

Vote
Current Rating
1
Rate Up
Rate Down
AleksThu, 25 Oct 2012 12:37:59 -0000

Awesome explanation. It vud b gud if u give some examples 2 d related topic. If so plz post me those. :-)

Rating
1
Rate Up
ShuvajitWed, 16 Jan 2013 20:49:44 -0000

In a plane if there are n points out of which m points are collinear, then The number of polygons with k sides that can be formed by joining them is nCk-mCk. But what if m<k?? also why dont we subtract those polygons that are formed (hypothetically) by k-1 collinear points and 1 non-collinear point; k-2 collinear points and 2 non-collinear points;…….3 collinear points and k-3 non-collinear points???

Vote
Current Rating
0
Rate Up
Rate Down
rizwanqadirmemonThu, 11 Oct 2012 16:20:03 -0000

In how many ways 16 dollars can be divided into 4 beggars and no any beggar got less than 3 dollars.

Rating
1
Rate Up
ShuvajitWed, 16 Jan 2013 20:58:19 -0000

set aside 4*3=12 dollars from 16 dollars. so we are left with 4 dollars to distribute among 4 beggars. partition theory gives 7C3 ways of doing this.

Vote
Current Rating
1
Rate Up
Rate Down
AnupCoolThu, 20 Sep 2012 19:00:31 -0000

These comments are really cool nd nice to revise

Vote
Current Rating
1
Rate Up
Rate Down
shiva235Fri, 24 Aug 2012 12:33:42 -0000

sir,what ever u kept before us will be very valuable….thank u

Vote
Current Rating
1
Rate Up
Rate Down
g0705Thu, 31 May 2012 19:45:34 -0000

how many five digit numbers can be formed using the digits 0,2,3,4,and 5 when repition is allowed such that the number formed is divisible by 2 or 5 or both..plz solve this and post the answer plzzz.

Rating
0
Rate Up
udayprince299Sat, 02 Jun 2012 13:50:27 -0000

numbers divisible by 2 are 1500
numbers divisible by 5 are 1000
numbers divisible by both 2 and 5 are 500

Vote
Current Rating
1
Rate Up
Rate Down
g0705Thu, 31 May 2012 19:24:49 -0000
can any one plz solve this and tell me how to do it… In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S? thanks in advance..

Rating
1
Rate Up
gnvSun, 01 Jul 2012 03:54:58 -0000
P and S can be arranged in 16 places;1 and 5 place,2 and 6 place,3 and 7 place ,4 and 8 place,5 and 9 place ,6 and 10 place,7 and 11 place,and 8 and 12 place since there should be 4 letters in between.Now we need to arrange remaining letters in between them.since there are 10 letters remaining ..10! ways is possible.but as there are 2 Ts repetition of counting is followed so divide it by 2! ….10*16
Rating
1
Rate Up
fazimohdWed, 19 Sep 2012 04:33:55 -0000

thr r 12 letrs in de word PERMUTATION. P nd S can b occupy in 1st nd 6th places/ 2nd nd 7th places/ 3rd nd 8th places/ 4th nd 9th/ 5th nd 10th/ 6th nd 11th/ 7th nd 12th places.. i.e., P nd S can b plcd in 7 ways. Also P nd S can be interchange their position. thus P nd S can be placed in 14 ways.
since positions of P nd s are fixed, d remaining ten places in 10*14= 25401600

Vote
Current Rating
1
Rate Up
Rate Down
reubenzibusSun, 12 Feb 2012 17:00:35 -0000

life has many struggles but only those that are determined win them… Mathematics needs determination beware!!!

Vote
Current Rating
1
Rate Up
Rate Down
MathsmadFri, 10 Feb 2012 05:17:50 -0000

hi….i hav a question sir. What r the types of questions r there in circular permutation? i need example for each type sir……..

Vote
Current Rating
1
Rate Up
Rate Down
mmnagpalMon, 23 Jan 2012 10:35:55 -0000

in short good knowledge to gain.

Vote
Current Rating
0
Rate Up
Rate Down
abhishekfigoMon, 22 Aug 2011 16:41:44 -0000

Vote
Current Rating
1
Rate Up
Rate Down
rnm_greenFri, 04 Mar 2011 18:24:50 -0000

nice representation

Vote
Current Rating
1
Rate Up
Rate Down
devspringSat, 14 Aug 2010 13:29:15 -0000

thanks buddy its add on to ma skills

Vote
Current Rating
1
Rate Up
Rate Down
haritham khanFri, 11 Jun 2010 12:51:29 -0000

although its a good rather best explanation but still iam not able to differentiate PC, so iam suggesting you to differentiate them through examples and figures, u must use same example and figures for both then it will be friutfull

Vote
Current Rating
1
Rate Up
Rate Down
amir saeedSun, 14 Mar 2010 12:12:10 -0000

please guide me how to solve this problem, if you joined all the vertices of heptagon, how many quadrilaterals will you get?

Rating
1
Rate Up
g0705Fri, 01 Jun 2012 06:11:46 -0000

its like choosins 4 vertices out of 7 vertices. coz quardilateral is of 4 vertices. now 7c4=7*4!=35..

Vote
Current Rating
1
Rate Up
Rate Down
itdoesntmatterSat, 09 Jan 2010 19:54:50 -0000

really nyc stuff

Vote
Current Rating
1
Rate Up
Rate Down
jatin luthraSat, 09 Jan 2010 18:47:39 -0000

thanx a lot
but
do u have similar notes for vectors n 3D too……………..

Vote
Current Rating
1
Rate Up
Rate Down
kunal jainSun, 13 Dec 2009 07:05:26 -0000

i like this alot

Vote
Current Rating
1
Rate Up
Rate Down
greentreeWed, 09 Dec 2009 12:48:09 -0000

not discriptive

Vote
Current Rating
1
Rate Up
Rate Down
praveen yadavSat, 17 Oct 2009 01:55:11 -0000

i think its enough to prepare this ………

Vote
Current Rating
1
Rate Up
Rate Down
Saurabh RoongtaSun, 20 Sep 2009 09:12:27 -0000

awsome stuff ………..

Vote
Current Rating
0
Rate Up
Rate Down
VedatmanThu, 04 Jun 2009 00:28:31 -0000

very good information……..thanks a lot sir

Vote
Current Rating
0
Rate Up
Rate Down
anirudh92Sun, 24 May 2009 01:56:05 -0000

very good

Vote
Current Rating
0
Rate Up
Rate Down
Tue, 19 May 2009 13:50:51 -0000

thanks

Vote
Current Rating
0
Rate Up
Rate Down
Tue, 19 May 2009 13:50:49 -0000

for someone like me who has v basic math knowledge examples of each type would be useful. few months ago you said it was due - has it been created? I think we often get these types of questions in gmat

Vote
Current Rating
0
Rate Up
Rate Down
suman sourabhSun, 19 Apr 2009 20:24:22 -0000

good job

Vote
Current Rating
0
Rate Up
Rate Down
gargi_lFri, 10 Apr 2009 13:27:07 -0000

Number of permutations of n different things, taken all at a time, when m specified things always come together is m

why not
m

Rating
0
Rate Up
Oren LahavFri, 10 Apr 2009 13:47:54 -0000

Say you have n items, and you're trying to organize them with m of the n items always coming together. (Clearly m < n). To organize the m items that come together, that's m!. Now to organize everything else, we have (n - m + 1)!, because we organize the items not in the M group, that's n - m, but we also have to put the M group together with them, so in total it comes to n - m + 1.

I hope that makes sense.

Vote
Current Rating
0
Rate Up
Rate Down

Textile is Enabled. View Reference.

### Apply to Top MBA Colleges accepting GMAT

Apply to Top MBA Colleges across the world accepting GMAT Scores