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Let ' $x$ ' be the number of ways in which $7$ identical balls can be coloured using $5$ different colours, no ball is supposed to be coloured with more than one colour and all the $5$ colours are to be used & ' $y$ ' be the number of ways in which the aforesaid work is done, when the balls are numbered.
Let " $a$ " be the number of ways in which $7$ distinct balls can be distributed into $5$ identical boxes so that no box is empty.
And " $b$ " be the number of ways in which $7$ distinct balls can be distributed into $5$ distinct boxes so that no box is empty, then which of the following is/are true?

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Important Questions on Permutation and Combination

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Let

A = {1, 2, 3, 4, 5, 6, 7, 8}, B = {9, 10, 11, 12, 13, 14, 15, 16}

and

C = {17, 18, 19, 20, 21, 22, 23, 24} .

Find the number of triples

(x, y, z)

such that

x \in A, y \in B, z \in C

and

x + y + z = 36

EASY

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

The number of ways in which

9

persons can be divided into three equal groups is

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

N

is the number of triangles of different shapes (i.e. not similar) whose angle are all integers (in degrees), what is

(\frac{N}{100}) ?

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

In an examination, there are

5

multiple choice questions with

3

choices, out of which exactly one is correct. There are

3

marks for each correct answer,

- 2

marks for each wrong answer and

0

mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets

5

marks is _____

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

The total number of

3 -

digit numbers whose sum of digits is

10

, is ..........

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

The number of ways to distribute

30

identical candies among four children

C_{1}, C_{2}, C_{3}

and

C_{4}

so that

C_{2}

receives atleast

4

and atmost

7

candies,

C_{3}

receives atleast

2

and atmost

6

candies, is equal to

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

The total number of positive integral solutions

(x, y, z)

such that

x y z = 24

is :

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Eight persons are to be transported from city

A

to city

B

in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

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_____

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is_____

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Let

A

be a matrix of order

2 \times 2

, whose entries are from the set

\{0, 1, 2, 3, 4, 5\}

. If the sum of all the entries of

A

is a prime number

p, 2 < p < 8

, then the number of such matrices

A

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Three schools send

2, 4

and

6

students, respectively, to a summer camp. The

12

students must be accommodated in

6

rooms numbered

1, 2, 3, 4, 5, 6

in such a way that each room has exactly

2

students and both from the same school. The number of ways, the students can be accommodated in the rooms is

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

The number of ways,

16

identical cubes, of which

11

are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least

2

blue cubes, is ______.

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Ari chooses

7

balls at random from

n

balls numbered

1

n .

If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen balls, find

n .

EASY

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Number of integral solutions to the equation

x + y + z = 21

, where

x \geq 1, y \geq 3, z \geq 4

, is equal to _____ .

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Let

X

be a set with exactly

5

elements and

Y

be a set with exactly

7

elements. If

α

is the number of one-one functions from

X \to Y

and

β

is the number of onto functions from

Y \to X

, then the value of

\frac{1}{5!} (β - α)

is _____-.

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

The number of triplets

(x, y, z)

where

x, y, z

are distinct non negative integers satisfying

x + y + z = 15

, is

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Consider the following statements
$i$ . Number of ways of placing $' n'$ objects in $k$ bins $k \leq n$ ) such that no bin is empty is ${}^{(n - 1)}C_{k - 1}$

$ii$ . Number of ways of writing a positive integer " $n$ ' into a sum of $k$ positive integers is ${}^{(n - 1)}C_{k - 1}$

$iii$ . Number of ways of placing ' $n$ ' objects in $k$ bins such that at least one bin is non-empty is ${}^{(n - 1)}C_{k - 1}$

$iv$ . ${}^{n}C_{k} - {}^{n - 1}C_{k} = {}^{(n - 1)}C_{k - 1}$

MEDIUM

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

Let

X

be a set of

5

elements. The number

d

of ordered pairs

(A, B)

of subsets of

X

such that

A \neq ϕ, B \neq ϕ, A \cap B = ϕ

satisfies

HARD

Mathematics>Algebra>Permutation and Combination>Division and Distribution of Objects

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices

(0, 0), (0, 41)

and

(41, 0)

Important Questions on Permutation and Combination

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