LFM Stats And Pure

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 2015 Q90 Binomial Distribution Identification and Application View

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is
(1) $22 \left( \frac { 1 } { 3 } \right) ^ { 11 }$
(2) $\frac { 5 } { 19 }$
(3) $55 \left( \frac { 2 } { 3 } \right) ^ { 10 }$
(4) $220 \left( \frac { 1 } { 3 } \right) ^ { 12 }$

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 2018 Q63 Forming Numbers with Digit Constraints View

$n$ - digit numbers are formed using only three digits 2,5 and 7 . The smallest value of $n$ for which 900 such distinct numbers can be formed, is
(1) 6
(2) 8
(3) 9
(4) 7

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 2019 Q69 Geometric Combinatorics View

Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is:
(1) 36
(2) 32
(3) 9
(4) 18

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 Q60 Subset Counting with Set-Theoretic Conditions View

Let $\cup _ { i = 1 } ^ { 50 } X _ { i } = \cup _ { i = 1 } ^ { n } Y _ { i } = T$, where each $X _ { i }$ contains 10 elements and each $Y _ { i }$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X _ { i }$'s and exactly 6 of sets $Y _ { i }$'s then $n$ is equal to:
(1) 15
(2) 50
(3) 45
(4) 30

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 Evaluation of a Finite or Infinite Sum View

If $n \geqslant 2$ is a positive integer, then the sum of the series ${ } ^ { n + 1 } C _ { 2 } + 2 \left( { } ^ { 2 } C _ { 2 } + { } ^ { 3 } C _ { 2 } + { } ^ { 4 } C _ { 2 } + \ldots + { } ^ { n } C _ { 2 } \right)$ is
(1) $\frac { n ( n - 1 ) ( 2 n + 1 ) } { 6 }$
(2) $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$
(3) $\frac { n ( n + 1 ) ^ { 2 } ( n + 2 ) } { 12 }$
(4) $\frac { n ( 2 n + 1 ) ( 3 n + 1 ) } { 6 }$

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 Q65 Factorial and Combinatorial Expression Simplification View

If ${ } ^ { n } P _ { r } = { } ^ { n } P _ { r + 1 }$ and ${ } ^ { n } C _ { r } = { } ^ { n } C _ { r - 1 }$, then the value of $r$ is equal to:
(1) 1
(2) 4
(3) 2
(4) 3

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

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12