[Combined PDF 4.0] - Permutations & Combinations.pdf

Permutations &
Combinations
11
CLASS
JEE 2024 Purnima Kaul
Addition & Multiplication
Principle

How many 3 digit numbers can be formed using the digits
1, 2 and 3 (without repetition).
EXAMPLE
How Many ?

How many 4 digit numbers can be formed using the digits
1, 2, 3 and 4 (without repetition).
EXAMPLE
How Many ?

Additionand
Multiplication
Principle

A person can travel from city A to city B via Road (Car/Bus/Bike),
Train (Express/Mail) or Flight (Economy/Business).
In how many ways can a person go from city A to city B ?

A person can travel from city A to city B via Road (Car/Bus/Bike),
Train (Express/Mail) or Flight (Economy/Business).
He can further travel from city B to city C via Road (Car/Bus/Bike) and
Train (Shatabdi/Express/Mail).
In how many ways can a person go from city A to city C via city B?

A stadium has 5 gates as shown. Find the number of ways of going in
and out as per following conditions.
1. Entry and exit from any door.
2. Entry and Exit from different doors.
3. Entry from side-1 and exit from side-2
4. Entry and exit should be from different sides.

Cities A, B, C and D are connected by a network of roads as shown
below. In how many ways can a person go from city A to city D
without visiting a city more than once?

Concept
Fillingthe vacant
spaces

How many 4 digit numbers can be formed using the digits 1, 2, 3, 4, 5
1. without repetition
2. Repetition allowed

How many 4 digit odd numbers can be formed using the digits
1, 2, 3, 4, 5 (without repetition)

How many 4 digit even numbers can be formed using the digits
1, 2, 3, 4, 5 (without repetition)

How many 4 digit numbers can be formed using the digits 0, 1, 2, 3, 4
1. without repetition
2. Repetition allowed

How many 4 digit numbers divisible by 4 can be formed using the
digits 0, 1, 2, 3, 4 (without repetition)

If the number of five digit numbers with distinct digits and 2 at
the 10th place is 336k, then k is equal to:
A. 4 B. 6 C. 7 D. 8
JEE Main 2020

In how many ways a student can answer a single option correct
question with four options?

In how many ways a student can answer a multiple correct question
with four options?

In how many ways can a student answer 10 question if
1. Questions are single correct type.
2. Questions are multiple correct type.

There are 10 lamps in a hall. Each one of them can be switched on
independently. The number of ways in which the hall can be
illuminated is
A. 102 B. 18 C. 210 D. 1023

JEE Main 2019
The number of 6 digit numbers that can be formed using the
digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is
repeated is:
A. 72 B. 60 C. 48 D. 36

Permutations &
Combinations
11
CLASS
Arrangement
Dictionary Problems

The number of 6 digit numbers that can be formed using the
digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is
repeated is:
A. 72 B. 60 C. 48 D. 36
JEE Main 2019

A. n+1Pr B. nPr C. r.nPr D. None of these
n-1Pr+ r . n-1Pr-1 =

The value of (2 . 1P0 - 3. 2P1 + 4 . 3P2 - …...up to 51th term) +
(1! - 2! + 3! -....up to 51th term) is equal to :
A. 1 - 51(51)! B. 1 + (51)! C. 1 + (52)! D. 1
JEE Main 2020

Number of ways of arranging ‘r’ objects out
of ‘n’ available distinct objects

Number of ways of arranging ‘ n ’ objects out
of ‘n’ available distinct objects

In a train 3 seats are vacant then in how many ways
can 5 passengers sit.

How many 3 digit numbers can be formed with only odd digits ?
(without repetition)

How many different words can be formed using all the letters of
the word DELHI. (words may be meaningless)

How many 9 and 4 letter words be created using the letters of the word
ALGORITHM (words may be meaningless)

How many numbers of five digits can be formed from the numbers
2, 0, 5, 3, 7 when repetition of digit is not allowed

Words are created by rearranging letters of the word TABLE and arranged
alphabetically. Then what is the position of the word TABLE?

The letters of the word RANDOM are written in all possible order and
these words are listed alphabetically. Then, find the rank of the word
RANDOM.

If the letters of the word 'MOTHER' be permuted and all the words so
formed (with or without meaning) be listed as in a dictionary, then the
position of the word 'MOTHER' is
JEE Main 2020

Permutations &
Combinations
11
CLASS
Arrangement of alike objects
Dictionary Problems

If all letters of word PURNIMA are arranged in dictionary order, find
dictionary rank of word PURNIMA :

How many 3 digit numbers can be formed using the digits 1, 2 and 2.
EXAMPLE:
Number of ways of arranging ‘(p+q+r)’ objects out of which ‘p’ are
alike of one kind and ‘q’ are alike of second kind and ‘r’ are distinct

EXAMPLE:
How many 4 digit numbers can be formed using the digits 1, 1, 1 and 2.
Number of ways of arranging ‘(p+q+r)’ objects out of which ‘p’ are
alike of one kind and ‘q’ are alike of second kind and ‘r’ are distinct

Number of ways of arranging ‘(p+q+r)’ objects out of which
‘p’ are alike of one kind and ‘q’ are alike of second kind and
‘r’ are distinct

How many words can be created by rearranging the letters of
the word MUMBAI?

How many words can be created by rearranging the letters of the
word ALLAHABAD?

If an insect wants to travel from point A to point B on the net via a
shortest possible path, then how many paths are possible?
A
B

If an insect wants to travel from point A to point B, via C on the net
using a shortest possible path, then how many paths are possible?
A
B

If an insect wants to travel from point A to point B, avoiding C on the net
using a shortest possible path, then how many paths are possible?
A
B

If all the words (with or without meaning) having five letters, formed
using the letters of the word SMALL and arranged as in a dictionary,
then the position of the word SMALL is :
A. 52nd B. 58th C. 46th D. 59th
JEE Main 2016

Words are created by rearranging letters of the word SUCCESS and
arranged alphabetically. Find the position of the word SUCCESS?

Permutations &
Combinations
11
CLASS
Selections

1. In how many ways 2 boys can be arranged out of a group of 3
boys.
2. In how many ways 2 boys can be selected out of a group of 3 boys.
EXAMPLE:
Number of ways of selecting ‘r’ objects out of ‘n’ available
distinct object

Number of ways of selecting ‘r’ objects out of ‘n’ available
distinct object

Relation between Permutation and Combination

In how many ways can a team of 4 boys and 4 girls be selected
out of 6 boys and 5 girls.

In how many ways can a team of 4 boys and 4 girls be selected out of
6 boys and 5 girls if a particular boy and a particular girl refuse to
work together.

A debate club consists of 6 girls and 4 boys. A team of 4 members
is to be selected from this club including the selection of a captain
(from among these 4 members) for the team. If the team has to
include at most one boy, then the number of ways of selecting the
team is
A. 380 B. 320 C. 260 D. 95
JEE Advanced 2016

A box contains 2 white balls, 3 black balls and 4 red balls. In how many
ways can three balls be drawn from the box if at least one black ball is to
be included in the draw?
(Balls of same colour are also distinct).

How many 3 digit numbers can be formed such that its digits are in
increasing A.P.

In a high school, a committee has to be formed from a group of 6 boys M1, M2,
M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.
I. Let α1 be the total number of ways in which the committee can be formed
such that the committee has 5 members, having exactly 3 boys and 2
girls.
II. Let α2 be the total number of ways in which the committee can be
formed such that the committee has at least 2 members, and having an
equal number of boys and girls.
III. Let α3 be the total number of ways in which the committee can be
formed such that the committee has 5 members, at least 2 of them being
girls.
IV. Let α4 be the total number of ways in which the committee can be
formed such that the committee has 4 members, having at least 2 girls
and such that both M₁ and G₁ are NOT in the committee together.
(JEE Adv. 2018)

A. P → 4; Q → 6; R → 2; S → 1 B. P → 1; Q → 4; R → 2; S → 3
C. P → 4; Q → 6; R → 5; S → 2 D. P → 4; Q → 2; R → 3; S → 1
List - 1 List - 2
P. The value of α1 is
Q. The value of α2 is
R. The value of α3 is
S. The value of α4 is
1. 136
2. 189
3. 192
4. 200
5. 381
6. 461

1) How many 4-lettered words can be formed using letters of word
ARTICLE?
2) How many 4-lettered words containing 2 vowels and 2 consonants?
3) How many words can be formed using all the letters when all vowels
together?
4) How many words can be formed using all the letters when all consonants
together?
5) How many words can be formed using all the letters when all vowels are
not together?
6) How many words can be formed starting with A?
7) How many words can be formed starting with E?
8) How many words can be formed starting with A and ending in E?
9) How many words can be formed maintaining the same relative order of
vowels?

1) How many 6-lettered words can be formed using letters of word
SECRET?
2) How many 6-lettered words starting with S?
3) How many 6-lettered words ending with E?
4) How many 6-lettered words starting with S and ending with E?
5) How many 6-lettered words starting with S or ending with E?
6) How many 6-lettered words which neither start with S nor end with E?
7) How many 6-lettered words with all Es together?
8) How many 6-lettered words where Es are not together?
9) How many 6-lettered words where S comes before E?

Permutations &
Combinations
11
CLASS
Geometry &
Chessboard Problems

Lines L1 and L2 are parallel to each other. 3 points are taken on L1 and
4 points are taken on L2. How many new straight lines can be drawn
passing through these points?

The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points
respectively on them. How many triangles can be constructed using
these interior points as vertices ?

The maximum number of intersection points of n circles and n straight
lines, among themselves is 80. Then value of n is:

There are 11 points on a plane of which 5 are concyclic. Other than these,
no other 4 points are concyclic. Then maximum number of circles that can
be drawn so that each contains at least three of the given points is:

Find number of diagonals in a hexagon.

Let n ≥ 2 be an integer. Take n distinct points on a circle and join
each pair of points by a line segment. Colour the line segment
joining every pair of adjacent points by blue and the rest by red.
If the number of red and blue line segments are equal, then the
value of n is
JEE Advanced 2016

Count the number of rectangles in a chess board.
B. C. D. none
A.

Count the number of squares in a chess board.
B. C. D. none
A.

Find the number of ways of keeping 2 identical rooks on an 8 ⨉ 8
chess board so that they are not in adjacent squares. (Two squares
are adjacent when they have a common side)

Permutations &
Combinations
11
CLASS
Bundle & Gap Method

BundleMethod
(Keepingthings
together)

A Library has 3 Mathematics, 2 Physics and 5 Chemistry books which
are to be arranged on a shelf. In how many ways these can be
arranged such that all books of same subject are together.
Considering:
1. Books of same subject are identical.
2. Books of same subject are also distinct.

arranged such that Mathematics books are together. Considering:

In how many ways the letters of the word IMAGINATION be
arranged such that
1. Both A’s are together
2. Both A’s, Both N’s and All three I’s are together
3. All vowels are together
4. All vowels are together and all consonants are together

Two families with three members each and one family with four members
are to be seated in a row. In how many ways can they be seated so that the
same family members are not separated?
A. 2!3!4! B. (3!)3.(4!) C. (3!)2.(4!) D. (3!).(4!)3
JEE Main 2020

GapMethod
(Keepingthings
separate)

arranged such that Mathematics books are separate. Considering:

In how many ways the letters of the word IMAGINATION be arranged such
that
1. Both A’s are separate
2. No two I’s are together
3. Two I’s are together but separated from third I.
4. Vowels and consonants are alternate

In how many ways 5 boys and 3 girls stand in a queue such that
1. No two girls are together.
2. Exactly two girls are together.

Let n be the number of ways in which 5 boys and 5 girls can stand in
a queue in such a way that all the girls stand consecutively in the
queue. Let m be the number of ways in which 5 boys and 5 girls can
stand in a queue in such a ways that exactly four girls stand
consecutively in the queue. Then the value of m/n is
JEE Advanced 2015

Permutations &
Combinations
11
CLASS
Circular Permutations

A. The circular permutations in which clockwise and the
anticlockwise arrangements give rise to different
permutations.
Example :
The number of ways in which n persons can be seated round a
table is (n-1)!
Seating arrangements of persons round a table.

arranging some beads to form a necklace.
B. The circular permutations in which clockwise and the
anticlockwise arrangements give same permutations.
Example :
The number of ways in which n different beads can be
arranged to form a necklace, is

In how many ways can 5 boys and 4 girls can sit on a circular table such
that
1. No two girls are together
2. All girls are together
3. Girls G1 and G2 sit together but not adjacent to G3.

How many different necklaces be made from 10 different beads such that
three particular beads always come together?

A. 6! 7! B. (6!)2 C. (7!)2 D. 7!
If seven women and seven men are to be seated around a circular
table such that there is a man on either side of every woman, then
the number of seating arrangements is -
JEE Main 2012

A. 5 x 6! B. 6 x 6! C. 7! D. 5 x 7!
The number of ways in which 5 boys and 3 girls can be seated on a
round table if a particular boy B1 and a particular girl G1 never sit
adjacent to each other, is :
JEE Main 2017

A. 360 B. 1440 C. 720 D. 2880
Eight people, including A and B, are to be seated around two identical
tables, each having a capacity of 4. The number of seating
arrangements, so that A and B are not at the same table, is

A. 7920 B. 2160 C. 10080 D. None
Eight people are to be seated, including A and B, four at a round table
and four at a straight table. The number of arrangements, in which A
and B are not at the round table, is

Five persons A, B, C, D and E are seated in a circular arrangement. If each
of them is given a hat of one of the three colours red, blue and green,
then the number of ways of distributing the hats such that the persons
seated in adjacent seats get different coloured hats is _____.
JEE Advanced 2019

Permutations &
Combinations
11
CLASS
Grouping/Partitioning

Number of ways of dividing 6 people into groups of
three, two and one.
Example :
Groups of unequal sizes

six and four.
Example :

5, 3 and 2.
Example :

Number of ways of dividing (m+n) distinct objects into two
groups of size m and n
1
Number of ways of dividing (m+n+p) distinct objects into three
groups of size m, n and p
2

2 and 2.
Groups of equal sizes
Example :

2, 2 and 2.
Example :

Number of ways of dividing 12 people into 2 groups of size
3 and 3 groups of size 2.
Groups of mixed sizes
Example :

Number of ways of dividing (2n) distinct objects into two
groups of size n each
Number of ways of dividing (3n) distinct objects into three
groups of size n each
1
2

Pay attention to DIVIDED v DISTRIBUTED

In how many ways can 8 distinct books be distributed among 2 students
in the ratio 1:3.

such that each student gets at least two books

Find the number of ways in which 14 people can depart in 2 cars of
capacity 4 and 2 autos of capacity 3, if two people insist on going in
the same car (internal arrangements to be ignored).

The set S = {1, 2, 3, ……………, 12} is to be partitioned into three sets A,
B, C of equal size. Thus, A ∪ B ∪ C = S, A ∩ B = B ∩ C = A ∩ C = Φ . the
number of ways to partition S is
A. B. D.
C.
AIEEE 2007

The total number of ways in which 5 balls of different colours can be
distributed among 3 persons so that each person gets at least one ball
is
A. 75 B. 150 C. 210 D. 243
JEE Advanced 2012

Permutations &
Combinations
11
CLASS
1. Selection of at least one
2. Number of Divisors

Selectionof
Objects
[Atleastone]

A basket contains 5 different fruits. In how many ways a person can select
atleast one fruit

Find the number of ways of selecting at least one fruit from 3 Mangoes,
4 apples and 5 Bananas

Number of ways of selecting at least one object out of 'n' available
objects of which 'p' are alike, 'q' are alike and rest are distinct
(p + 1)(q + 1)(2n-p-q) - 1
Number of ways of selecting at least one object out of 'n' available
distinct objects (2n - 1)
Number of ways of selecting at least one object out of (p+q+r)
available objects of which 'p' are alike, 'q' are alike and 'r' are alike.
(p + 1)(q + 1)(r + 1) - 1
1
2
3
Selection of at least one

A room has 10 bulbs which can be switched on independently. In how
many ways can the room be illuminated ?

Applicationof
conceptofSelection:
NumberofDivisors

Let N = p1
𝛂1 .p2
𝛂2 . p3
𝛂3 ……. pk
𝛂k , where p1, p2, p3, ……. pk are
different primes and 𝛂1, 𝛂2, 𝛂3, ………. 𝛂k are natural numbers then:
The total number of divisors of N including 1 and N is:
(𝛂1 + 1) (𝛂2 + 1)(𝛂3 + 1) ……… (𝛂k + 1)
1.
Number of Divisors

For the number N = 2700, Find total number of divisors.

For the number N = 2700, Find total number of even and odd divisors.

For the number N = 2700, Find the number of divisors divisible by
15 but not by 4.

Total number of divisors of n = 25 . 34 . 510 . 76 that are of the
form 4λ + 2, λ≥ 1 is equal to :
A. 385 B. 55 C. 384 D. 54

The total number of divisors of N excluding 1 and N is:
(𝛂1 + 1) (𝛂2 + 1)(𝛂3 + 1) ……… (𝛂k + 1) - 2
2.
Let N = p1
𝛂1 .p2
𝛂2 . p3
𝛂3 ……. pk
Number of Proper Divisors

The sum of these divisors is = (p1
0 + p1
1 + p1
2 + ……. + p1
𝛂1)
(p2
0 + p2
1 + p2
2 + ……. + p2
𝛂2 ) ….. (pk
0 + pk
1 + pk
2 + ……. + pk
𝛂k)
3.
Let N = p1
𝛂1 .p2
𝛂2 . p3
𝛂3 ……. pk
Sum of Divisors

For the number N = 2700, Find the sum of divisors.

For the number N = 2700, Find the sum of even and odd divisors

Sum of all divisors of 5400 whose unit digit is 0, is :
A. 5400 B. 10800 C. 16800 D. 14400

If the number of ways of selecting 3 numbers out of 1, 2, 3, …, 2n + 1
numbers such that they are in A.P. is 441, then the sum of divisors
of n is :
A. 21 B. 22 C. 32 D. None of these

The number of ways in which N can be resolved as a product
of two factors is
4.
Let N = p1
𝛂1 .p2
𝛂2 . p3
𝛂3 ……. pk
Resolving into product of two numbers

Find the number of ways in which 12 can be resolved into two factors.

Find the number of ways in which 36 can be resolved into two factors.

5.
Let N = p1
𝛂1 .p2
𝛂2 . p3
𝛂3 ……. pk
Resolving into product of two co-prime numbers

Number of ways in which 38808 can be expressed as a product of
two co-prime factors are :
A. 11 B. 8 C. 16 D. 2

Permutations &
Combinations
11
CLASS
1. Partition/Beggar’s Method
2. Number of Solution of Equation

Distributionof
Distinctobjects
(withoutrestriction)

Find the number of ways of distributing 5 different ice creams among 3
students. (without any restrictions)

Distribution of distinct objects with restrictions is done through
division into groups.
such that each student gets at least two books.
Note:
Example

Distributionof
likeobjects
(withoutrestriction)

Here we arrange like objects in a row and imagine a partition in
between to generate required conditions.
Distributing 5 like balls into 2 people
Partition Method or Ball and Stick Method
Example

Partition Method or Ball and Stick Method
Example

Beggar’s Method Shortcut
Example
Example

Find the number of ways of distributing 10 mangoes among 3 students, without any restriction. (here
all mangoes are alike).
A. 10C3 B. 12C2 C. 10P3 D. (10!) - (3!)

Find the number of ways of distributing 10 mangoes, 6 apples and 4 bananas among 3
students such that every student gets at least one mango.
A. 22C2 B. 19C2 C. (9C2)(8C2)(6C2) D. (12C2)(8C2)(6C2) - 1

Applicationof partition
method
(Numberof solutionof
Equation)

Find the number of non negative integral solutions of the equation
x + y + z = 10

Find the number of positive integral solutions of the equation
x + y + z = 10

Find the number of ways in which 30 identical things are distributed
among 6 persons if each gets (i) odd number of things (ii) even number of
things.

Find the number of integral solutions of the equation x + y + z = 10
satisfying the condition; x >-1, y≥ -1, z ≥ 2

Find the number of non negative integral solutions of the
inequality x + y + z ≤ 10

Find the number of non negative integral solutions of the
inequality
(i) 7< x + y + z ≤ 10
(i) 7< x + y + z ≤ 10
(iii) 7< x + y + z < 10
(iv) 7< x + y + z < 10

The number of non negative integer solutions of x + y + 2z = 20 is
A. 81 B. 125
C.
112 D.
121

In how many ways can a 12 step staircase be climbed taking 1 step or 2
steps at a time?

Permutations &
Combinations
11
CLASS
Inclusion Exclusion Principle
Derangement Problem

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
Understanding IEP for Two Sets

n(A ∪ B ∪ C) = [n(A) + n(B) + n (C)] - [n(A ∩ B) + n(B ∩ C) + n(C ∩ A)]
+ [n(A ∩ B ∩ C)]
Understanding IEP for Two Sets

Find the number of positive integers from 1 to 100, which are
divisible by at least one of 3 or 5.

neither divisible by 3 nor by 5.

neither divisible by 2 nor by 3 nor by 5.
A ∩ B : Divisible by 2 and 3 ⇒ Divisible by 6
B ∩ C : Divisible by 3 and 5 ⇒ Divisible by 15
C ∩ A : Divisible by 5 and 2 ⇒ Divisible by 10
A ∩ B ∩ C : Divisible by 2, 3 and 5 ⇒ Divisible by 30
A : Divisible by 2
B : Divisible by 3
C : Divisible by 5

Find the number of ways of dealing a five card hand from a regular 52
card deck such that the hand contains at least one card of each suit.
Total possible ways
C1: cards given from 3 suits (1 suit missing)
C2: cards given from 2 suits (2 suits missing)
C3: cards given from 1 suit (3 suits missing)

Derangement
(Specialcase
of I.E.P.)

It is the number of ways of arranging ‘n’ distinct objects in
‘n’ places such that no object goes to its designated place.
Derangement

Find the number of ways of placing 5 letters in 5 envelopes such that
no letter goes in the correct envelope.

Find the number of ways of placing 5 letters in 5 envelopes such that
no letter goes in the correct envelope.
Total possible ways
C1: 1 letter in correct envelope

D(1) 0
D(2) 1
D(3) 2
D(4) 9
D(5) 44
D(6) 265
Important Values:
Derangement

264
A. 265
B.
JEE Adv. 2014
53
C. 67
D.
Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are
to be placed in envelopes so that each envelope contains exactly one
card and no card is placed in the envelope bearing the same number
and moreover the card numbered 1 is always placed in envelope
numbered 2. Then the number of ways it can be done is

Find the sum of all the 3 digit numbers that can be formed using the
digits 1, 2 and 3

((n -1)!) x ( sum of digits ) ⨉ ( 1 + 10 + 100 + …….)
For non zero Distinct Digits

digits 1, 2, 3 and 4

digits 0, 1, 2, and 3.

Find the sum of all the 4 digit numbers that can be formed using the digits
1, 1, 2, and 3.

Let p be a prime number n be a positive integer, then
exponent of p in n! Is denoted by Ep(n!) and is given by
Exponent of Prime

Find the exponent of 2 in 10!.

The number of zeros at the end of 108! is
A. 10 B. 13 C. 25 D. 26

The exponent of 7 in 100C50
A. 0 B. 1 C. 2 D. 3

Selectionof
Non-Consecutive
Objects

In how many ways can 4 trees be cut from 10 trees in a row such
that no two trees cut are consecutive.
Method-1: Gap Method

Method-2: Using Equation solving/Beggars’ Method

Find the number of ways of selecting 3 people out of 10 people sitting on
a round table such that no two people selected are consecutive.
A. 42
B. 50
C. 56
D. 62

An engineer is required to visit a factory for exactly four days during the
first 15 days of every month and it is mandatory that no two visits take
place on consecutive days. Then the number of all possible ways in
which such visits to the factory can be made by the engineer during 1-15
June 2021 is
JEE Adv. 2020

Form a 4-lettered word from INDEPENDENCE.

Form a 4-lettered word from HIPPOPOTAMUSES.

11
CLASS
Permutations &
Combinations
Multinomial Theorem

In this theorem, we try to write all possible outcomes in powers
of a random variable ‘x’ and then calculate coefficient of required power.
Multinomial Theorem

The Coefficient of xr in the expansion of (1-x)-n is n+r-1Cr
Important Results

Find the number of ways of distributing 10 mangoes among 3 students,
without any restriction. (here all mangoes are alike).
A. 10C3 B. 12C2 C. 10P3 D. (10!) - (3!)

If a dice is rolled three times, then find the number of ways of getting sum
equal to 10.
A. 27 B. 36 C. 45 D. None

The number of ways of selecting 8 fruits from unlimited supply of
Mangoes, Apples, Bananas and Oranges is:
165
B.
A. 81 C. 38 D. 27C8

A. 45 B. 46 C. 48 D. 50
The number of terms in the expansion of (x + y + 2z)8 is

The number of distinct terms in the expansion of
(x + 2y - 3z + 5w - 7u)n is
A. n + 1 B. n+4C4
C. n+5C5
D.

Selectionof
Objects
(ExactNumber)

Find the number of ways of selecting exactly 4 fruits out of 6 Apples,
5 Mangoes and 4 Bananas

Find the number of 4 letter words that can be created using the
letters of the word MATHEMATICS.

[Combined PDF 4.0] - Permutations & Combinations.pdf

[Combined PDF 4.0] - Permutations & Combinations.pdf

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