LFM Stats And Pure

jee-main 2014 Q63 Selection with Group/Category Constraints View

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by 66 , then the number of men who participated in the tournament lies in the interval
(1) $( 11,13 ]$
(2) $( 14,17 )$
(3) $[ 10,12 )$
(4) $[ 8,9 ]$

jee-main 2015 Q65 Subset Counting with Set-Theoretic Conditions View

Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets of the set $A \times B$, each having at least three elements is
(1) 510
(2) 219
(3) 256
(4) 275

jee-main 2016 Q63 Basic Combination Computation View

If $\frac { { } ^ { n + 2 } C _ { 6 } } { { } ^ { n - 2 } P _ { 2 } } = 11$, then $n$ satisfies the equation:
(1) $n ^ { 2 } + n - 110 = 0$
(2) $n ^ { 2 } + 2 n - 80 = 0$
(3) $n ^ { 2 } + 3 n - 108 = 0$
(4) $n ^ { 2 } + 5 n - 84 = 0$

jee-main 2017 Q63 Selection with Group/Category Constraints View

A man $X$ has 7 friends, 4 of them are ladies and 3 are men. His wife $Y$ also has 7 friends, 3 of them are ladies and 4 are men. Assume $X$ and $Y$ have no common friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $X$ and $Y$ are in this party is:
(1) 485
(2) 468
(3) 469
(4) 484

jee-main 2019 Q63 Combinatorial Identity or Bijection Proof View

If $\sum _ { r = 0 } ^ { 25 } \left\{ \left( { } ^ { 50 } C _ { r } \right) \left( { } ^ { 50 - r } C _ { 25 - r } \right) \right\} = K \left( { } ^ { 50 } C _ { 25 } \right)$, then $K$ is equal to
(1) $2 ^ { 25 }$
(2) $2 ^ { 25 } - 1$
(3) $2 ^ { 24 }$
(4) $( 25 ) ^ { 2 }$

jee-main 2019 Q63 Subset Counting with Set-Theoretic Conditions View

Let $S = \{ 1,2,3 , \ldots , 100 \}$, then number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :
(1) $2 ^ { 100 } - 1$
(2) $2 ^ { 50 } + 1$
(3) $2 ^ { 50 } \left( 2 ^ { 50 } - 1 \right)$
(4) $2 ^ { 50 } - 1$

jee-main 2019 Q63 Selection with Group/Category Constraints View

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to
(1) 24
(2) 27
(3) 25
(4) 28

jee-main 2019 Q63 Selection with Group/Category Constraints View

A committee of 11 member is to be formed from 8 males and 5 females. If $m$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then:
(1) $m = n = 68$
(2) $n = m - 8$
(3) $m = n = 78$
(4) $m + n = 68$

jee-main 2019 Q64 Basic Combination Computation View

Consider three boxes, each containing 10 balls labelled $1,2 , \ldots , 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n _ { i }$, the label of the ball drawn from the $i ^ { \text {th } }$ box, $( i = 1,2,3 )$. Then, the number of ways in which the balls can be chosen such that $n _ { 1 } < n _ { 2 } < n _ { 3 }$ is:
(1) 240
(2) 82
(3) 120
(4) 164

jee-main 2020 Q53 Geometric Combinatorics View

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of $n$ is
(1) 201
(2) 200
(3) 101
(4) 199

jee-main 2020 Q53 Selection with Group/Category Constraints View

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:
(1) 3000
(2) 1500
(3) 2255
(4) 2250

jee-main 2020 Q53 Basic Combination Computation View

If $a , b$ and $c$ are the greatest values of ${}^{ 19 } C _ { p } , {}^{ 20 } C _ { q }$ and ${}^{ 21 } C _ { r }$ respectively, then:
(1) $\frac { a } { 11 } = \frac { b } { 22 } = \frac { c } { 21 }$
(2) $\frac { a } { 10 } = \frac { b } { 20 } = \frac { c } { 21 }$
(3) $\frac { a } { 11 } = \frac { b } { 22 } = \frac { c } { 42 }$
(4) $\frac { a } { 10 } = \frac { b } { 11 } = \frac { c } { 42 }$

jee-main 2020 Q55 Combinatorial Identity or Bijection Proof View

The value of $\sum _ { r = 0 } ^ { 20 } { } ^ { 50 - r } C _ { 6 }$ is equal to:
(1) ${ } ^ { 51 } C _ { 7 } - { } ^ { 30 } C _ { 7 }$
(2) ${ } ^ { 50 } C _ { 7 } - { } ^ { 30 } C _ { 7 }$
(3) ${ } ^ { 50 } C _ { 6 } - { } ^ { 30 } C _ { 6 }$
(4) ${ } ^ { 51 } C _ { 7 } + { } ^ { 30 } C _ { 7 }$

jee-main 2020 Q56 Basic Combination Computation View

The number of ordered pairs $( r , k )$ for which $6 . { } ^ { 35 } C _ { r } = \left( k ^ { 2 } - 3 \right) . { } ^ { 36 } C _ { r + 1 }$, where $k$ is an integer is
(1) 3
(2) 2
(3) 6
(4) 4

jee-main 2020 Q71 Selection with Group/Category Constraints View

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is $\_\_\_\_$

jee-main 2020 Q71 Selection with Group/Category Constraints View

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

jee-main 2021 Q62 Selection with Group/Category Constraints View

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:
(1) 1050
(2) 1625
(3) 575
(4) 560

jee-main 2021 Q62 Geometric Combinatorics View

Consider a rectangle $ABCD$ having $5,6,7,9$ points in the interior of the line segments $AB , BC , CD , DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta - \alpha)$ is equal to
(1) 795
(2) 1173
(3) 1890
(4) 717

jee-main 2021 Q62 Geometric Combinatorics View

Let $P _ { 1 } , \quad P _ { 2 } \ldots , \quad P _ { 15 }$ be 15 points on a circle. The number of distinct triangles formed by points $P _ { i } , \quad P _ { j } , \quad P _ { k }$ such that $i + j + k \neq 15$, is :
(1) 455
(2) 419
(3) 12
(4) 443

jee-main 2021 Q62 Geometric Combinatorics View

If the sides $A B , B C$ and $C A$ of a triangle $A B C$ have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:
(1) 364
(2) 240
(3) 333
(4) 360

jee-main 2021 Q63 Selection with Group/Category Constraints View

Team '$A$' consists of 7 boys and $n$ girls and Team '$B$' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to:
(1) 5
(2) 2
(3) 4
(4) 6

jee-main 2021 Q68 Subset Counting with Set-Theoretic Conditions View

Let $Z$ be the set of all integers, $A = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ $B = \left\{ ( x , y ) \in Z \times Z : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ and $C = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \leq 4 \right\}$ If the total number of relations from $A \cap B$ to $A \cap C$ is $2 ^ { p }$, then the value of $p$ is: (1) 25 (2) 9 (3) 16 (4) 49

jee-main 2021 Q76 Combinatorial Identity or Bijection Proof View

If $\sum _ { k = 1 } ^ { 10 } K ^ { 2 } \left( { } ^ { 10 } C _ { K } \right) ^ { 2 } = 22000 L$, then $L$ is equal to

jee-main 2021 Q82 Counting Arrangements with Run or Pattern Constraints View

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is

jee-main 2021 Q82 Selection with Arithmetic or Divisibility Conditions View

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

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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