Core Syllabus Module

Permutations and Combinations

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Main 2021 (27 Aug Shift 2)

LEVELJEE Advanced

Let $S = {1, 2, 3, 4, 5, 6, 9}$ . Then the number of elements in the set $T = {A \subseteq S : A \neq = ϕ and the sum of all the elements of A is not a multiple of 3}$ is .

JEE Main 2026 (21 January Shift 1)

LEVELJEE Advanced

The number of strictly increasing functions $f$ from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq = i$ for $1 \leq i \leq 6$ , is equal to :

JEE Main 2021 (01 Sep Shift 2)

LEVELJEE Advanced

All the arrangements, with or without meaning, of the word $F A R M E R$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $F A R M E R$ in this list is .

JEE Main 2022 (25 July Shift 1)

LEVELJEE Advanced

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ______.

JEE Main 2023 (25 January Shift 1)

LEVELJEE Advanced

Let $S = {1, 2, 3, 5, 7, 10, 11}$ . The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3, is ____.

JEE Main 2024 (09 Apr Shift 2)

LEVELJEE Advanced

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is _________

JEE Advanced 2018

LEVELJEE Advanced

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 ________.

JEE Advanced 2014

LEVELJEE Main

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

A group of 9 students $s_{1}, s_{2}, \dots, s_{9}$ is to be divided to form three teams $X, Y$ and $Z$ of sizes 2, 3 and 4 respectively. Suppose that $s_{1}$ cannot be selected for the team $X$ , and $s_{2}$ cannot be selected for team $Y$ . Then the number of ways to form such teams, is ________.

JEE Advanced 2022

LEVELJEE Main

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?

JEE Main 2026 (22 January Shift 1)

LEVELJEE Main

Let ABC be a triangle. Consider four points $p_{1}, p_{2}, p_{3}, p_{4}$ on the side AB, five points $p_{5}, p_{6}, p_{7}, p_{8}, p_{9}$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_{1}, p_{2}, \dots, p_{13}$ , is ......... .

JEE Main 2022 (27 June Shift 1)

LEVELJEE Main

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 ______.

JEE Main 2022 (28 June Shift 2)

LEVELJEE Main

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

Comprehension Passage

Let

a_{n}

denote the number of all

n

-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let

b_{n}

= the number of such

n

-digit integers ending with digit 1 and

c_{n}

= the number of such

n

-digit integers ending with digit 0.

Question 1:

The value of $b_{6}$ is

Which of the following is correct?

(A)

a_{17} = a_{16} + a_{15}

(B)

c_{17} \neq = c_{16} + c_{15}

(C)

b_{17} \neq = b_{16} + c_{16}

(D)

a_{17} = c_{17} + b_{16}

JEE Main 2022 (29 July Shift 2)

LEVELJEE Main

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is ______.

JEE Main 2026 (24 January Shift 2)

LEVELJEE Main

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is

JEE Main 2023 (30 January Shift 1)

LEVELJEE Main

Let $S = {1, 2, 3, 4, 5, 6}$ . Then the number of one-one functions $f : S \to P (S)$ , where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is

JEE Main 2023 (31 January Shift 2)

LEVELJEE Main

Let $A = [a_{ij}], a_{ij} \in Z \cap [0, 4], 1 \leq i, j \leq 2$ . The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is ______.

JEE Main 2025 April

LEVELJEE Main

Consider the sets $A = {(x, y) \in R \times R : x^{2} + y^{2} = 25}$ , $B = {(x, y) \in R \times R : x^{2} + 9 y^{2} = 144}$ , $C = {(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4}$ , and $D = A \cap B$ . The total number of one-one functions from the set D to the set C is:

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 way that exactly four girls stand consecutively in the queue. Then the value of $\frac{m}{n}$ is

Micro-Topic Strategy Matrix

Fundamental Principle of Counting

Factorial Notation

Linear Permutations

Circular Permutations

Combinations and Selection

Formation of Groups

Derangement Principle

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={1,2,3,4,5,6,9}. Then the number of elements in the set T={A⊆S:A=ϕ and the sum of all the elements of A is not a multiple of 3} is .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of strictly increasing functions f from the set {1,2,3,4,5,6} to the set {1,2,3,…,9} such that f(i)=i for 1≤i≤6, is equal to :

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={1,2,3,5,7,10,11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is ____.

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is _________

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?

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 ______.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of b6​ is

Which of the following is correct?

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is ______.

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={1,2,3,4,5,6}. Then the number of one-one functions f:S→P(S), where P(S) denote the power set of S, such that f(n)⊂f(m) where n<m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=[aij​],aij​∈Z∩[0,4],1≤i,j≤2. The number of matrices A such that the sum of all entries is a prime number p∈(2,13) is ______.

Let $S = {1, 2, 3, 4, 5, 6, 9}$ . Then the number of elements in the set $T = {A \subseteq S : A \neq = ϕ and the sum of all the elements of A is not a multiple of 3}$ is .

The number of strictly increasing functions $f$ from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq = i$ for $1 \leq i \leq 6$ , is equal to :

Let $S = {1, 2, 3, 5, 7, 10, 11}$ . The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3, is ____.

The value of $b_{6}$ is

Let $S = {1, 2, 3, 4, 5, 6}$ . Then the number of one-one functions $f : S \to P (S)$ , where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is

Let $A = [a_{ij}], a_{ij} \in Z \cap [0, 4], 1 \leq i, j \leq 2$ . The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is ______.