Permutations & Combinations

24 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q9·MCQ·Moderate
Open
Counting strictly increasing functions
The number of strictly increasing functions $f$ from the set $\{1, 2, 3, 4, 5, 6\}$ to the set $\{1, 2, 3, \ldots, 9\}$ such that $f(i) \neq i$ for $1 \le i \le 6$ , is equal to:
1. (1)21
2. (2)27
3. (3)22
4. (4)28
We corrected a printing error in the original paper
Source printed the second set as “ $(1, 2, 3, \ldots, 9)$ ” (parentheses instead of set braces).
MathsJEE Main 21 Jan 2026, Shift 1·Q23·Numerical·Moderate
Open
Counting ordered pairs
Let $S = \{(m, n) : m, n \in \{1, 2, 3, \ldots, 50\}\}$ . If the number of elements $(m, n)$ in $S$ such that $6^m + 9^n$ is a multiple of 5 is $p$ and the number of elements $(m, n)$ in $S$ such that $m + n$ is a square of a prime number is $q$ , then $p + q$ is equal to .
MathsJEE Main 21 Jan 2026, Shift 2·Q20·MCQ·Easy–moderate
Open
Exponent of prime in factorial
The largest $n \in \mathbb{N}$ , for which $7^n$ divides $101!$ , is:
1. (1)16
2. (2)18
3. (3)15
4. (4)19
MathsJEE Main 22 Jan 2026, Shift 1·Q25·Numerical·Moderate
Open
Selection of points on triangle sides
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 $p_1$ , $p_2$ , $\ldots$ , $p_{13}$ , is .
MathsJEE Main 22 Jan 2026, Shift 2·Q25·Numerical·Easy–moderate
Open
Counting subsets
Let $S$ be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \ge 2 \text{ and the product of all elements of } B \text{ is even}\}$ is .
MathsJEE Main 23 Jan 2026, Shift 1·Q25·Numerical·Moderate
Open
Arrangements with repeated letters
The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is .
MathsJEE Main 23 Jan 2026, Shift 2·Q15·MCQ·Easy–moderate
Open
Distribution of identical objects
The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is
1. (1)429
2. (2)384
3. (3)403
4. (4)455
MathsJEE Main 23 Jan 2026, Shift 2·Q25·Numerical·Moderate
Open
Counting numbers by digits
Let $S$ denote the set of 4-digit numbers $abcd$ such that $a > b > c > d$ and $P$ denote the set of 5-digit numbers having product of its digits equal to 20. Then $n(S) + n(P)$ is equal to .
MathsJEE Main 24 Jan 2026, Shift 1·Q23·Numerical·Moderate
Open
Counting numbers with divisibility
The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is .
MathsJEE Main 24 Jan 2026, Shift 2·Q9·MCQ·Easy–moderate
Open
Rank of a word in dictionary
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:
1. (1)1580
2. (2)1578
3. (3)1579
4. (4)1581
MathsJEE Main 24 Jan 2026, Shift 2·Q10·MCQ·Easy–moderate
Open
Exponent of a prime in factorial
The largest value of $n$ , for which $40^n$ divides $60!$ , is
1. (1)13
2. (2)11
3. (3)12
4. (4)14
MathsJEE Main 28 Jan 2026, Shift 1·Q7·MCQ·Moderate
Open
Arrangements with repeated digits
Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ . Let $x$ be the number of 9-digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9-digit numbers formed using the digits of the set $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then,
1. (1) $29x = 5y$
2. (2) $45x = 7y$
3. (3) $21x = 4y$
4. (4) $56x = 9y$
MathsJEE Main 28 Jan 2026, Shift 2·Q22·Numerical·Easy–moderate
Open
Arrangements of persons
Three persons enter in a lift at the ground floor. The lift will go upto $10^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to .
MathsJEE Main 2 Apr 2026, Shift 1·Q6·MCQ·Moderate
Open
Arrangements with each digit used
The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 5 and 7 such that each digit is used at least once, is;
1. (1)15400
2. (2)17800
3. (3)16800
4. (4)29400
MathsJEE Main 2 Apr 2026, Shift 2·Q6·MCQ·Easy–moderate
Open
Properties of binomial coefficients
If for $3 \le r \le 30$ , $\left({}^{30}C_{30-r}\right) + 3\left({}^{30}C_{31-r}\right) + 3\left({}^{30}C_{32-r}\right) + \left({}^{30}C_{33-r}\right) = {}^{m}C_{r},$ then $m$ equals:
1. (1)31
2. (2)32
3. (3)33
4. (4)34
MathsJEE Main 2 Apr 2026, Shift 2·Q7·MCQ·Easy–moderate
Open
Triangles from polygon vertices
Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$ -sided regular polygon. If $p_{n+1} - p_n = 66$ , then the sum of all distinct prime divisors of $n$ is:
1. (1)7
2. (2)8
3. (3)5
4. (4)6
We corrected a printing error in the original paper
Source printed “an $n$ -side regular polygon”.
MathsJEE Main 4 Apr 2026, Shift 1·Q8·MCQ·Easy–moderate
Open
Arrangements with restrictions
The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:
1. (1)5040
2. (2)3050
3. (3)3410
4. (4)4320
MathsJEE Main 4 Apr 2026, Shift 2·Q17·MCQ·Easy–moderate
Open
Non-negative integer solutions
Let $A = \{(a, b, c) : a, b, c$ are non-negative integers and $a + b + 2c = 22\}$ . The $n(A)$ is equal to:
1. (1)121
2. (2)124
3. (3)144
4. (4)169
MathsJEE Main 5 Apr 2026, Shift 1·Q21·Numerical·Moderate
Open
Counting one-one functions
Let $A = \{1, 2, 3, 4, 5, 6\}$ . The number of one-one functions $f : A \to A$ such that $f(1) \ge 3$ , $f(3) \le 4$ and $f(2) + f(3) = 5$ , is .
MathsJEE Main 5 Apr 2026, Shift 1·Q22·Numerical·Moderate
Open
Counting game sequences
Two players $A$ and $B$ play a series of games of badminton. The player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player $A$ wins the series is .
MathsJEE Main 5 Apr 2026, Shift 2·Q9·MCQ·Easy–moderate
Open
Selection with restrictions
A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is:
1. (1)4100
2. (2)4140
3. (3)4230
4. (4)4290
MathsJEE Main 6 Apr 2026, Shift 1·Q7·MCQ·Easy–moderate
Open
Selection and arrangement of letters
The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:
1. (1)2670
2. (2)2840
3. (3)2920
4. (4)3600
MathsJEE Main 6 Apr 2026, Shift 2·Q7·MCQ·Moderate
Open
Distribution into groups and floors
A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the $10^{\text{th}}$ floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to:
1. (1)2184
2. (2)3064
3. (3)7056
4. (4)11340
MathsJEE Main 8 Apr 2026, Shift 2·Q9·MCQ·Easy–moderate
Open
Distribution into non-empty groups
A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is
1. (1)18
2. (2)36
3. (3)39
4. (4)72