PERMUTATIONS & COMBINATIONS

Fundamental principle of Counting:
We have two principles of counting the “Sum rule” and the “Product rule”.
Event A can happen in = m ways.
Event B can happen in = n ways.
Then, the chances of either (A or B) happening are given by ⎯⎯⎯⎯⎯→ can occur in m + n ways (Sum Rule)
The chances of both (A and B) happening simultaneously are given by ⎯⎯⎯⎯⎯→ can occur in m × n ways (Product
Rule)
Concept of nPr and nCr:
r
nP → Number of arrangements of n different objects taken ‘r’ at a time
nCr → Number of selections of n different objects taken ‘r‘ at a time.
Formulae of Permutation:
1. Permutations of n different things taken 'r' at a time is denoted by nPr and is given by
nPr =
(n r)!
n!
−
2. The total number of arrangements of n things taken ‘r’ at a time, in which a particular thing always
occurs = n–1Pr–1.
E.g. The number of ways in a basketball game in which 5 out of 8 selected players can play at different
positions, such that the captain always plays at the centre position = 8 – 1 P5 – 1 = 7P4 = 210.
3. The total number of permutations of n different things taken ‘r’ at a time in which a particular thing never
occurs = n–1Pr.
E.g. The number of ways in which we can form a 4 letter word from the letters of the word COMBINE
such that the word never contains B = 7–1P4 = 6P4 = 30
4. The number of arrangements when things are not all different such as arrangement of n things, when p
of them are of one kind, q of another kind, r is still of another kind and so on, the total number of
permutations is given by
(p! q! r !.......)
n! .
E.g. The total arrangements of the letters of the word “M A T H E M A T I C S” in which M, A and T are
repeated twice respectively =
2!2!2!
11!
5. The number of permutations of n different things taking ‘r’ at a time when each thing may be repeated
any number of times in any permutations is given by (n × n × n × n × n……………..r times) i.e. nr ways.

E.g. The total numbers of ways in which 7 balls can be distributed amongst 9 persons (when any man
can get any number of balls) = 97 ways.
Circular Permutations:
In linear permutation, we fill first place by n ways and next in (n – 1) ways and so on, but in circular
arrangement we don’t have any first place. So fix any object as a first place and arrange the rest (n – 1)
objects around it. Hence, we have to arrange 1 less than the total number of things.
i. Number of circular permutations of n things all taken at a time = (n – 1)!
ii. Number of circular permutations of n different things taking ‘r’ at a time =
r
Pr
n
.
iii. If there is no difference between clockwise and anticlockwise arrangements, the total number of circular
permutations of n things taking all at a time is
2
(n −1)!
& the total number of circular permutations n
when taking ‘r’ at a time all will be
2r
Pr
n
.
Formulae of Combination:
1. Number of combinations of n dissimilar things taken 'r' at a time is denoted by nCr & is given by
nCr =
(n r)!r!
n!
−
2. Number of combinations of n different things taken ‘r’ at a time in which ‘p’ particular things will always
occur is n–pCr–p
E.g. The number of ways a basketball team of 5 players chosen from 8 players, so that the captain be
included in the team = 8 – 1 C5 – 1 = 7C4 = 35
3. Number of combinations of n dissimilar things taken 'r' at a time in which 'p' particular things will never
occur is n–pCr
E.g. The number of ways a basketball team of 5 players chosen out of 10 players, such that the player
named Saurav should not be included in the team = 10 – 1 C5 = 9C5 = 126.
4. The number of ways in which (m + n) things can be divided into two groups containing m & n things
respectively (m + n)Cn =
m! n!
(m + n)!
= (m + n)Cm

5. If 2m things are to be divided into two groups, each containing m things, the number of ways =
[2(m! ) ]
(2m)!
2 .
6. The number of ways to divide n things into different groups, one containing p things, another q things &
so on is equal to
p!.q!.r!......
(p + q + r + .....)!
Where {n = p + q + r +…}
Distribution of Identical Objects:
The total number of ways of dividing n identical items among r persons, each of whom can receive 0, 1, 2, or
more items (≤ n) is n + r – 1Cr – 1.
OR
The total number of ways of dividing n identical objects into r groups, if blank groups are allowed, is
n + r – 1Cr – 1.
Example:
How many non – negative integral solutions are possible for the given equation?
x + y + z = 16
Hint: Here, if we look out to the problem we will find 16 objects have to be distributed among 3 different
persons (i.e. x, y, z).
Hence n = 16, r = 3 and total number of non – negative solutions = 3 1
16 3 1C −
+ − = 2
18C = 153.
Some important Results:
i. Number of lines with n points = nC2.
Q For making a line exactly two points are required. So the number of ways in which we can choose
two points out of n point is nC2.
Combination is used here because a line from A to B is the same as from B to A. So AB & BA are the
same.
(i) n lines can intersect at a maximum of nC2 points.
(ii) Number of triangles with n points = nC3.
(iii) Number of diagonals in n sided polygon = nC2 – n
ii. 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
m!
1
(n! )
(mn)!
m ⎟
⎟
⎠

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DETAILED SYLLABUS FOR NIMCET

DISTRIBUTION OF QUESTIONS:
The question paper will contain 120 multiple choice questions covering the following topics:
1. Mathematics 40 questions
2. Analytical Ability & Logical Reasoning 50 questions
3. Computer Awareness 15 questions
4. General English 15 questions

DETAILED SYLLABUS
MATHEMATICS : (40 questions)
Set Theory: Concepts of sets – Union – Intersection – Cardinality – Elementary counting
permutations and combinations

Probability and Statistics: Basic concepts of probability theory, Averages, Dependent and
independent events, frequency distributions, measures of central tendencies and dispersions.

Algebra: Fundamental operations in Algebra, Expansions, Factorization, simultaneous
linear / quadratic equations, indices, logarithms, arithmetic, geometric and harmonic
progressions, determinants and matrices.

Coordinate Geometry: Rectangular Cartesian coordinates, distance formulae, equation of a
line, intersection of lines, pair of straight lines, equations of a circle, parabola, ellipse, and
hyperbola.

Calculus: Limit of functions, Continuous function, Differentiation of function, tangents and
normals, simple examples of maxima and minima. Integration of function by parts, by
substitution and by partial fraction; Definite integrals, Applications of Definite Integrals to
areas.

Vectors: Position vector, addition and subtraction of vectors, scalar and vector products and
their applications to simple geometrical problems and mechanics.

Trigonometry: Simple identities, trigonometric equations, properties of triangles, solution of
triangles, heights and distances, General solutions of trigonometric equations.

ANALYTICAL ABILITY AND LOGICAL REASONING : (50 questions)
The questions in this section will cover logical reasoning and quantitative aptitude. Some of
the questions will be on comprehension of a logical situation and questions based on the facts
given in the passage.


SYLLABUS CONTINUATION
COMPUTER AWARENESS : (15 questions)
Computer Basics: Organization of a Computer, Central Processing Unit (CPU), Structure of
instructions in CPU, input/output devices, computer memory, back-up devices.
Data Representation: Representation of characters, integers and fractions, binary and
hexadecimal representations, Binary Arithmetic: Addition, subtraction, multiplication,
division, simple arithmetic and two’s complement arithmetic, floating point representation of
numbers, Boolean algebra, truth tables, Venn diagrams.

GENERAL ENGLISH : (15 questions)
Questions in this section will be designed to test the candidates’ general understanding of the
English language. There will be questions on the following topics.
Comprehension, Vocabulary, Basic English Grammar (like usage of correct forms of verbs,
prepositions and articles). Word power, Synonyms and Antonyms, Meanings of words and
phrases, Technical writing.

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TEST YOUR G.K.

1. Who created the character, James Bond?
(A) Ian Buster
(B) Ian Fleming
(C) Ian Bosewell
(D) None of these

2. Hindi movie Mughal-e-Azam was directed by
(A) K – Asif
(B) Abdul Kareem
(C) Mehboob
(D) Nadia brothers

3. The first Indian talkie movie was
(A) Jahanara
(B) Alamara
(C) Aag
(D) Khayal

4. The title “Historic Man” was given to
(A) Jalauddin
(B) Sohrab Modi
(C) Mohd Yasin
(D) Mirza Younis

5. The first ever Indian (from Bollywood) to be given a chance for Hollywood movie
(A) Devanand
(B) Raj Kapoor
(C) I.S. Johar
(D) Dilip Kumar

6. The movie based on the life of Haji Mastan is
(A) Dewaar
(B) Don
(C) Nehle Pe Dehla
(D) Heera Panna

7. 872 runs scored in a day, was ODI played between
(A) Australia – England
(B) Australia – South Africa
(C) England – South Africa
(D) South Africa – New Zealand

8. The name “Baichung Bhutia” is associated with
(A) Football
(B) Hockey
(C) Polo (
) Derby

9. Which country bagged maximum medals in the 18th Commonwealth Games?
(A) China
(B) England
(C) New Zealand
(D) Australia

10. The first ever Test victory in Pakistan against the hosts was under the captainship of
(A) Saurav Ganguly
(B) Rahul Dravid
(C) Kapil Dev (
D) Mohd. Azharuddin

11. The Football World Cup final 1998 was played between
(A) France and Brazil
(B) Brazil and Germany
(C) Brazil and Spain
(D) Italy and Brazil

12. Graham Thorpe is a well known name in the game of
(A) Table tennis
(B) Lawn tennis
(C) Baseball
(D) None of these

13. The Indian cricketer addressed as Jumbo is
(A) Anil Kumble
(B) Zaheer Khan
(C) Ramesh Powar
(D) Yuvraj Singh

14. “Floats like a butterfly, stings like a bee” is a title given to
(A) Mohd Ali
(B) Bruce Lee
(C) Mike Tyson
(D) Graham Thorpe

15. Which country has bagged football World Cup four times?
(A) Brazil
(B) France
(C) Italy
(D) None of these

16. The name “Pulela Gopichand” is associated with
(A) Lawn tennis
(B) Table tennis
(C) Kabaddi
(D) Badminton
17. Geet Sethi is a former _________ champion.
(A) Badminton
(B) Football
(C) Billiards
(D) Chess

18. Name the country that introduced bodline bowling in cricket.
(A) Australia
(B) West Indies
(C) South Africa
(D) New Zealand
19. Wrestling show WWF is now known as
(A) WWS
(B) WWM
(C) WWE
(D) WWP

20. The name Kerry Packer is associated with
(A) Cricket
(B) Football
(C) Basketball
(D) Volleyball

21. The term Rabbit is associated with
(A) Cricket (B) Swimming (C) Golf (D) Volley ball
22. Renowned sitar player Ravi Shanker’s debut as a Music director was for the movie
(A) Kasam
(B) Krishna
(C) Payal
(D) Meera

23. The title “Little Master” goes with
(A) Sunil Gavaskar
(B) Sachin Tendulkar
(C) Brian Lara
(D) Alan Border

24. Greg Norman signature championship is associated with
(A) Chess
(B) Cricket
(C) Golf
(D) Snooker

25. Name the movie in which Amitabh Bachan played a triple role
(A) Mahaan
(B) Kasuti
(C) Sate Pe Satta
(D) Raste Ka Pathar

26. The movie based on 1962 Indo-China war is
(A) Hindustan Ki Kasam
(B) Akraman
(C) Des Pardes
(D) Haqueekat

27. Rishi Kapoor for the first time on the Cinema screen, which movie we are talking of
(A) Bobby
(B) Rafoo Chakkar
(C) Mera Nam Joker
(D) Kalia

28. Shiv Kumar Sharma is well known for his performance in
(A) Flute
(B) Piano
(C) Sitar
(D) Santoor
29. “Dadra” is
(A) Railway station
(B) A village in M.P.
(C) Raga
(D) Musical instrument







ANSWERS
1. (B) 2. (A) 3. (B) 4. (B) 5. (C) 6. (A) 7. (B) 8. (A)
9. (D) 10. (B) 11. (A) 12. (D) 13. (A) 14. (A) 15. (A) 16. (D)
17. (C) 18. (A) 19. (C) 20. (A) 21. (A) 22. (D) 23. (A) 24. (C)
25. (A) 26. (D) 27. (C) 28. (D) 29. (C)