LFM Stats And Pure

isi-entrance 2016 Q25 4 marks Subset Counting with Set-Theoretic Conditions View

Let $S = \{ 1, 2, \ldots , n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$

isi-entrance 2016 Q26 4 marks Counting Functions or Mappings with Constraints View

The number of maps $f$ from the set $\{ 1, 2, 3 \}$ into the set $\{ 1, 2, 3, 4, 5 \}$ such that $f(i) \leq f(j)$ whenever $i < j$ is
(A) 60
(B) 50
(C) 35
(D) 30

isi-entrance 2016 Q26 4 marks Counting Functions or Mappings with Constraints View

The number of maps $f$ from the set $\{ 1, 2, 3 \}$ into the set $\{ 1, 2, 3, 4, 5 \}$ such that $f ( i ) \leq f ( j )$ whenever $i < j$ is
(A) 60
(B) 50
(C) 35
(D) 30

isi-entrance 2016 Q27 4 marks Counting Functions or Mappings with Constraints View

Consider three boxes, each containing 10 balls labelled $1, 2, \ldots, 10$. Suppose one ball is drawn from each of the boxes. Denote by $n_i$, the label of the ball drawn from the $i$-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
(A) 120
(B) 130
(C) 150
(D) 160

isi-entrance 2016 Q27 4 marks Counting Functions or Mappings with Constraints View

Consider three boxes, each containing 10 balls labelled $1, 2, \ldots, 10$. Suppose one ball is drawn from each of the boxes. Denote by $n _ { i }$, the label of the ball drawn from the $i$-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
(A) 120
(B) 130
(C) 150
(D) 160

isi-entrance 2016 Q40 4 marks Selection with Group/Category Constraints View

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom{10}{3} \binom{7}{4} 2^4$
(B) $\binom{10}{3} \binom{7}{4}$
(C) $\binom{10}{3} 2^7$
(D) $\binom{10}{3} \binom{14}{4}$

isi-entrance 2016 Q40 4 marks Selection with Group/Category Constraints View

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$

isi-entrance 2016 Q59 4 marks Selection with Adjacency or Spacing Constraints View

The number of ways in which one can select six distinct integers from the set $\{1, 2, 3, \cdots, 49\}$, such that no two consecutive integers are selected, is
(A) $\binom{49}{6} - 5\binom{48}{5}$
(B) $\binom{43}{6}$
(C) $\binom{25}{6}$
(D) $\binom{44}{6}$

isi-entrance 2016 Q59 4 marks Selection with Adjacency or Spacing Constraints View

The number of ways in which one can select six distinct integers from the set $\{ 1, 2, 3, \cdots, 49 \}$, such that no two consecutive integers are selected, is
(A) $\binom { 49 } { 6 } - 5 \binom { 48 } { 5 }$
(B) $\binom { 43 } { 6 }$
(C) $\binom { 25 } { 6 }$
(D) $\binom { 44 } { 6 }$

isi-entrance 2017 Q10 Combinatorial Probability View

Let $V$ be the set of vertices of a regular polygon with twenty sides. Three distinct vertices are chosen at random from $V$. Then, the probability that the chosen triplet are the vertices of a right angled triangle is
(A) $\frac{7}{19}$
(B) $\frac{3}{19}$
(C) $\frac{3}{38}$
(D) $\frac{1}{38}$.

isi-entrance 2018 Q2 Combinatorial Probability View

An office has 8 officers including two who are twins. Two teams, Red and Blue, of 4 officers each are to be formed randomly. What is the probability that the twins would be together in the Red team?
(A) $\frac { 1 } { 6 }$
(B) $\frac { 3 } { 7 }$
(C) $\frac { 1 } { 4 }$
(D) $\frac { 3 } { 14 }$

isi-entrance 2018 Q9 Lattice Path Counting View

An up-right path is a sequence of points $\mathbf { a } _ { 0 } = \left( x _ { 0 } , y _ { 0 } \right) , \mathbf { a } _ { 1 } = \left( x _ { 1 } , y _ { 1 } \right) , \mathbf { a } _ { 2 } = ( x _ { 2 } , y _ { 2 } ), \ldots$ such that $\mathbf { a } _ { i + 1 } - \mathbf { a } _ { i }$ is either $( 1,0 )$ or $( 0,1 )$. The number of up-right paths from $( 0,0 )$ to $( 100,100 )$ which pass through $( 1,2 )$ is:
(A) $3 \cdot \binom { 197 } { 99 }$
(B) $3 \cdot \binom { 100 } { 50 }$
(C) $2 \cdot \binom { 197 } { 98 }$
(D) $3 \cdot \binom { 197 } { 100 }$.

isi-entrance 2019 Q16 Distribution of Objects into Bins/Groups View

A school allowed the students of a class to go to swim during the days March 11th to March 15, 2019. The minimum number of students the class should have had that ensures that at least two of them went to swim on the same set of dates is:
(A) 6
(B) 32
(C) 33
(D) 121 .

isi-entrance 2020 Q1 Subset Counting with Set-Theoretic Conditions View

The number of subsets of $\{ 1,2,3 , \ldots , 10 \}$ having an odd number of elements is
(A) 1024
(B) 512
(C) 256
(D) 50 .

isi-entrance 2020 Q6 Partitioning into Teams or Groups View

A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?
(A) $\frac { 64 ! } { 32 ! 2 ^ { 32 } }$
(B) $\binom { 64 } { 2 } \binom { 62 } { 2 } \cdots \binom { 4 } { 2 } \binom { 2 } { 2 }$
(C) $\frac { 64 ! } { 32 ! 32 ! }$
(D) $\frac { 64 ! } { 2 ^ { 64 } }$

isi-entrance 2020 Q15 Counting Functions or Mappings with Constraints View

Let $A = \left\{ x _ { 1 } , x _ { 2 } , \ldots , x _ { 50 } \right\}$ and $B = \left\{ y _ { 1 } , y _ { 2 } , \ldots , y _ { 20 } \right\}$ be two sets of real numbers. What is the total number of functions $f : A \rightarrow B$ such that $f$ is onto and $f \left( x _ { 1 } \right) \leq f \left( x _ { 2 } \right) \leq \cdots \leq f \left( x _ { 50 } \right)$ ?
(A) $\binom { 49 } { 19 }$
(B) $\binom { 49 } { 20 }$
(C) $\binom { 50 } { 19 }$
(D) $\binom { 50 } { 20 }$

isi-entrance 2021 Q11 Combinatorial Probability View

A box has 13 distinct pairs of socks. Let $p _ { r }$ denote the probability of having at least one matching pair among a bunch of $r$ socks drawn at random from the box. If $r _ { 0 }$ is the maximum possible value of $r$ such that $p _ { r } < 1$, then the value of $p _ { r _ { 0 } }$ is
(A) $1 - \frac { 12 } { { } ^ { 26 } C _ { 12 } }$.
(B) $1 - \frac { 13 } { { } ^ { 26 } C _ { 13 } }$.
(C) $1 - \frac { 2 ^ { 13 } } { { } ^ { 26 } C _ { 13 } }$.
(D) $1 - \frac { 2 ^ { 12 } } { { } ^ { 26 } C _ { 12 } }$.

isi-entrance 2022 Q8 Counting Functions or Mappings with Constraints View

Let $( n _ { 1 } , n _ { 2 } , \cdots , n _ { 12 } )$ be a permutation of the numbers $1,2 , \cdots , 12$. The number of arrangements with $$n _ { 1 } > n _ { 2 } > n _ { 3 } > n _ { 4 } > n _ { 5 } > n _ { 6 }$$ and $$n _ { 6 } < n _ { 7 } < n _ { 8 } < n _ { 9 } < n _ { 10 } < n _ { 11 } < n _ { 12 }$$ equals:
(A) $\binom { 12 } { 5 }$
(B) $\binom { 12 } { 6 }$
(C) $\binom { 11 } { 6 }$
(D) $\frac { 11 ! } { 2 }$

isi-entrance 2022 Q10 Counting or Combinatorial Problems on APs View

In how many ways can we choose $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ from the set $\{ 1,2 , \ldots , 30 \}$ such that $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in arithmetic progression?
(A) 135
(B) 145
(C) 155
(D) 165

isi-entrance 2023 Q30 Counting Functions or Mappings with Constraints View

How many functions $f : \{ 1,2 , \ldots , 10 \} \rightarrow \{ 1 , \ldots , 2000 \}$, which satisfy $$f ( i + 1 ) - f ( i ) \geq 20 , \text { for all } 1 \leq i \leq 9 ,$$ are there?
(A) $10 ! \binom { 1829 } { 10 }$
(B) $11 ! \binom { 1830 } { 11 }$
(C) $\binom { 1829 } { 10 }$
(D) $\binom { 1830 } { 11 }$

isi-entrance 2024 Q12 Distribution of Objects to Positions or Containers View

Suppose 40 distinguishable balls are to be distributed into 4 different boxes such that each box gets exactly 10 balls. Out of these 40 balls, 10 are defective and 30 are non-defective. In how many ways can the balls be distributed such that all the defective balls go to the first two boxes?
(A) $\frac{40!}{(10!)^4}$
(B) $\frac{30! \cdot 20!}{(10!)^5}$
(C) $\frac{20! \cdot 20!}{(10!)^5}$
(D) $\frac{30! \cdot 10!}{(10!)^4}$

isi-entrance 2024 Q18 Counting Functions with Composition or Mapping Constraints View

Let $A = \{1, \ldots, 5\}$ and $B = \{1, \ldots, 10\}$. Then the number of ordered pairs $(f, g)$ of functions $f : A \rightarrow B$ and $g : B \rightarrow A$ satisfying $(g \circ f)(a) = a$ for all $a \in A$ is
(A) $\frac{10!}{5!} \times 5^5$
(B) $5^{10} \times 5!$
(C) $10! \times 5!$
(D) $\binom{10}{5} \times 10^5$

isi-entrance 2026 QB6 Subset Counting with Set-Theoretic Conditions View

Let $X$ be the set $\{ 1,2,3 , \ldots , 10 \}$ and $P$ the subset $\{ 1,2,3,4,5 \}$. The number of subsets $Q$ of $X$ such that $P \cap Q = \{ 3 \}$ is
(A) 1
(B) $2 ^ { 4 }$
(C) $2 ^ { 5 }$
(D) $2 ^ { 9 }$

isi-entrance 2026 Q8 Selection with Group/Category Constraints View

A box contains 10 red cards numbered $1 , \ldots , 10$ and 10 black cards numbered $1 , \ldots , 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(a) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$.
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$.
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$.
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$.

italy-esame-di-stato 2024 Q2 Combinatorial Probability View

2. In a piggy bank there are 15 coins, of which 9 are 1 euro coins and the other 6 are 2 euro coins. 6 coins are drawn simultaneously. – What is the probability that the total value of the coins drawn is exactly 10 euros? – What is the probability that the total value of the coins drawn is at most 10 euros?

1
2
3
4
5
6
7
8
9
10
11

Load questions...

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