LFM Stats And Pure

jee-main 2022 Q63 Combinatorial Identity or Bijection Proof View

If $\sum_{k=1}^{31} \left({}^{31}\mathrm{C}_k\right)\left({}^{31}\mathrm{C}_{k-1}\right) - \sum_{k=1}^{30}\left({}^{30}\mathrm{C}_k\right)\left({}^{30}\mathrm{C}_{k-1}\right) = \frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha \in R$, then the value of $16\alpha$ is equal to
(1) 1411
(2) 1320
(3) 1615
(4) 1855

jee-main 2022 Q73 Counting Functions or Mappings with Constraints View

The number of bijective functions $f:\{1,3,5,7,\cdots,99\} \rightarrow \{2,4,6,8,\cdots,100\}$ if $f(3) > f(5) > f(7) \cdots > f(99)$ is
(1) ${}^{50}C_1$
(2) ${}^{50}C_2$
(3) $\frac{50!}{2}$
(4) ${}^{50}C_3 \times 3!$

jee-main 2022 Q82 Selection with Group/Category Constraints View

There are ten boys $B _ { 1 } , B _ { 2 } , \ldots , B _ { 10 }$ and five girls $G _ { 1 } , G _ { 2 } , \ldots G _ { 5 }$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B _ { 1 }$ and $B _ { 2 }$ together should not be the members of a group, is $\_\_\_\_$.

jee-main 2022 Q82 Selection with Group/Category Constraints View

A class contains $b$ boys and $g$ girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then $b + 3g$ is equal to $\_\_\_\_$.

jee-main 2023 Q62 Partitioning into Teams or Groups View

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
(1) 1120
(2) 3360
(3) 1680
(4) 560

jee-main 2023 Q63 Basic Combination Computation View

If all the six digit numbers $\mathrm { x } _ { 1 } \mathrm { x } _ { 2 } \mathrm { x } _ { 3 } \mathrm { x } _ { 4 } \mathrm { x } _ { 5 } \mathrm { x } _ { 6 }$ with $0 < \mathrm { x } _ { 1 } < \mathrm { x } _ { 2 } < \mathrm { x } _ { 3 } < \mathrm { x } _ { 4 } < \mathrm { x } _ { 5 } < \mathrm { x } _ { 6 }$ are arranged in the increasing order, then the sum of the digits in the $72 ^ { \text {th} }$ number is $\_\_\_\_$ .

jee-main 2023 Q64 Selection with Group/Category Constraints View

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is $\_\_\_\_$.

jee-main 2023 Q64 Selection with Group/Category Constraints View

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is $\_\_\_\_$.

jee-main 2023 Q64 Basic Combination Computation View

Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is

jee-main 2023 Q67 Combinatorial Identity or Bijection Proof View

$\sum _ { k = 0 } ^ { 6 } { } ^ { 51 - k } C _ { 3 }$ is equal to
(1) ${ } ^ { 51 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(2) ${ } ^ { 51 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$
(3) ${ } ^ { 52 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(4) ${ } ^ { 52 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$

jee-main 2023 Q75 Subset Counting with Set-Theoretic Conditions View

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is
(1) 752
(2) 782
(3) 792
(4) 772

jee-main 2024 Q62 Geometric Combinatorics View

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
(1) 48
(2) 56
(3) 24
(4) 16

jee-main 2024 Q62 Selection with Group/Category Constraints View

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to : (1) 179 (2) 177 (3) 181 (4) 175

jee-main 2024 Q63 Counting Integer Solutions to Equations View

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136

jee-main 2024 Q65 Basic Combination Computation View

If for some $m, n$; ${}^{6}C_m + 2\,{}^{6}C_{m+1} + {}^{6}C_{m+2} > {}^{8}C_3$ and ${}^{n-1}P_3 : {}^{n}P_4 = 1 : 8$, then ${}^{n}P_{m+1} + {}^{n+1}C_m$ is equal to
(1) 380
(2) 376
(3) 384
(4) 372

jee-main 2024 Q65 Subset Counting with Set-Theoretic Conditions View

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $\mathrm { P } ( \mathrm { m } , \mathrm { n } )$ from the point $\mathrm { Q } ( - 2 , - 3 )$ is
(1) 10
(2) 6
(3) 4
(4) 8

jee-main 2024 Q68 Partitioning into Teams or Groups View

Let the set $S = \{ 2,4,8,16 , \ldots , 512 \}$ be partitioned into 3 sets $A , B , C$ with equal number of elements such that $\mathrm { A } \cup \mathrm { B } \cup \mathrm { C } = \mathrm { S }$ and $\mathrm { A } \cap \mathrm { B } = \mathrm { B } \cap \mathrm { C } = \mathrm { A } \cap \mathrm { C } = \phi$. The maximum number of such possible partitions of $S$ is equal to:
(1) 1680
(2) 1640
(3) 1520
(4) 1710

jee-main 2024 Q81 Geometric Combinatorics View

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n = 1, 2, 3, \ldots, 10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to:

jee-main 2024 Q81 Selection with Group/Category Constraints View

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is $\_\_\_\_$

jee-main 2024 Q82 Selection with Group/Category Constraints View

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections: A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is $\underline{\hspace{1cm}}$.

jee-main 2024 Q87 Counting Functions or Mappings with Constraints View

Let $A = \{ ( x , y ) : 2 x + 3 y = 23 , x , y \in \mathbb { N } \}$ and $B = \{ x : ( x , y ) \in A \}$. Then the number of one-one functions from $A$ to $B$ is equal to $\_\_\_\_$

jee-main 2025 Q1 Selection with Group/Category Constraints View

Group A consists of 7 boys and 3 girls, while group $B$ consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to :
(1) 8750
(2) 9100
(3) 8925
(4) 8575

jee-main 2025 Q5 Basic Combination Computation View

Let ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } - 1 } = 28 , { } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } } = 56$ and ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } + 1 } = 70$. Let $\mathrm { A } ( 4 \cos t , 4 \sin t ) , \mathrm { B } ( 2 \sin t , - 2 \cos \mathrm { t } )$ and $C \left( 3 r - n , r ^ { 2 } - n - 1 \right)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $( 3 x - 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = \alpha$, is the locus of the centroid of triangle ABC, then $\alpha$ equals
(1) 6
(2) 18
(3) 8
(4) 20

jee-main 2025 Q10 Selection with Adjacency or Spacing Constraints View

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:
(1) 5148
(2) 6084
(3) 4356
(4) 14950

kyotsu-test 2011 QCourse1-II-Q1 Distribution of Objects to Positions or Containers View

There are six boxes numbered from 1 to 6. We are to put four balls of different sizes into these boxes.
(1) There are altogether $\mathbf{AA}$ ways to put the four balls into the boxes.
(2) There are $\mathbf{CDE}$ ways to put the four balls into four separate boxes.
(3) There are $\mathbf{FGH}$ ways to put three balls into one box and the fourth ball into another.
(4) There are $\mathbf{IJK}$ ways to put at least one ball into the box numbered 1.

1
2
3
4
5
6
7
8
9
10

Load questions...

LFM Stats And Pure

Completing the square and sketching 61

Inequalities 181

Discriminant and conditions for roots 65

Solving quadratics and applications 131

Curve Sketching 228

Factor & Remainder Theorem 61

Polynomial Division & Manipulation 35

Binomial Theorem (positive integer n) 197

Function Transformations 28

Composite & Inverse Functions 211

Partial Fractions 12

Generalised Binomial Theorem 10

Complex Numbers Arithmetic 207

Complex Numbers Argand & Loci 159

Permutations & Arrangements 255

Combinations & Selection 237

Measures of Location and Spread 181

Probability Definitions 212

Tree Diagrams 18

Principle of Inclusion/Exclusion 33

Independent Events 35

Conditional Probability 167

Discrete Probability Distributions 176

Uniform Distribution 3

Binomial Distribution 101

Geometric Distribution 8

Hypergeometric Distribution 9

Negative Binomial Distribution 1

Modelling and Hypothesis Testing 22

Hypothesis test of binomial distributions 2

Data representation 21

Continuous Probability Distributions and Random Variables 322

Continuous Uniform Random Variables 3

Geometric Probability 16

Normal Distribution 87

Approximating Binomial to Normal Distribution 4

Sign Change & Interval Methods 26

Fixed Point Iteration 9

Newton-Raphson method 6