LFM Stats And Pure

grandes-ecoles 2023 Q1 Group Order and Structure Theorems View

Recall the cardinality of $\mathcal{S}_n$. Deduce that $R \geq 1$.

grandes-ecoles 2023 Q2 Combinatorial Identity or Bijection Proof View

For $k \in \llbracket 0, n \rrbracket$, show that the number of permutations of $\llbracket 1, n \rrbracket$ having exactly $k$ fixed points is $\binom{n}{k} d_{n-k}$.
Deduce that $P_n\left(X_n = k\right) = \frac{d_{n-k}}{k!(n-k)!}$.

grandes-ecoles 2024 Q3 Combinatorial Structures on Permutation Matrices/Groups View

Establish that $$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = -1\right\}$$ and deduce the probability that a permutation of $\mathfrak{S}_{n}$ drawn uniformly at random has a prescribed signature.

grandes-ecoles 2024 Q4 Combinatorial Structures on Permutation Matrices/Groups View

For $\sigma \in \mathfrak{S}_{n}$, specify the condition on $\nu(\sigma)$ for which $\sigma \in \mathfrak{D}_{n}$. Deduce that $$\operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = -1\right\} + (-1)^{n-1}(n-1).$$

iran-konkur 2014 Q154 Distribution of Objects into Bins/Groups View

154- The number of ordered triples, with non-negative integer and non-positive integer coordinates, such that the sum of every three coordinates of each set equals $10$ and each coordinate is less than $6$, is which of the following?
$$17 \ (1) \hspace{2cm} 18 \ (2) \hspace{2cm} 20 \ (3) \hspace{2cm} 21 \ (4)$$

iran-konkur 2017 Q154 Probability via Permutation Counting View

154- Six numbered balls are randomly placed in 3 boxes. What is the probability that no box remains empty?
(1) $\dfrac{5}{14}$ (2) $\dfrac{5}{12}$ (3) $\dfrac{3}{7}$ (4) $\dfrac{7}{12}$

iran-konkur 2019 Q148 Counting Functions with Constraints View

148. The number of surjective (onto) functions from a set with 6 elements to a set with 3 elements is which of the following?
(1) $360$ (2) $450$ (3) $480$ (4) $540$

iran-konkur 2020 Q142 Forming Numbers with Digit Constraints View

142- How many four-digit natural numbers are divisible by 5, with non-repeating digits?
\[ (1)\quad 948 \qquad (2)\quad 952 \qquad (3)\quad 968 \qquad (4)\quad 972 \]

iran-konkur 2020 Q148 Linear Arrangement with Constraints View

148 -- In a competition, 3 drivers participate for three consecutive days with 3 cars on routes A, B, and C. Each driver selects only one route and one car per day, and the scheduling is done in the form of a Latin square. In how many ways can the scheduling be done such that on the first day, no one selects car A?

2	3	1
3	1	2
1	2	3

[(1)] $1$
[(2)] $2$
[(3)] $3$
[(4)] $4$

iran-konkur 2021 Q137 Distribution of Objects into Bins/Groups View

137. The number of correct non-negative integer solutions of the equation $x_1 + x_2 + x_3 = \dfrac{10}{x_4}$ is which of the following?
(1) $60$ (2) $72$ (3) $81$ (4) $96$

iran-konkur 2022 Q125 Forming Numbers with Digit Constraints View

125-- How many five-digit natural numbers can be written with non-repeating digits such that among those digits, one is between two even and two odd digits?
(1) $1850$ (2) $1950$ (3) $2150$ (4) $2500$

iran-konkur 2022 Q149 Forming Numbers with Digit Constraints View

149- How many natural numbers less than $6000$ have digit sum $8$?
(4) $168$ (3) $164$ (2) $165$ (1) $155$

iran-konkur 2023 Q21 Linear Arrangement with Constraints View

21 -- In how many ways can 4 ministers, each with one assistant, sit in two rows of 8 seats facing each other so that each minister sits exactly opposite their own assistant?
(1) $24$ (2) $32$ (3) $48$ (4) $64$

isi-entrance 2009 Q8 Counting Integer Solutions to Equations View

Let $f(n)$ be the number of ways to write a positive integer as an ordered sum of three non-negative integers, where each integer is chosen from $\{0, 1, 2, \ldots, 2n-1\}$ (i.e., using $n$ colours with values $0$ to $2n-1$). Find $f(n)$.

isi-entrance 2010 Q18 Lattice Path / Grid Route Counting View

A person $X$ standing at a point $P$ on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps $X$ comes to the original position $P$. Then the number of distinct paths that $X$ can take is
(a) 196
(b) 256
(c) 344
(d) 400

isi-entrance 2011 Q7 Circular Arrangement View

In how many ways can 3 couples sit around a round table such that men and women alternate and none of the couples sit together?
(a) 1
(b) 2
(c) $5! / 3$
(d) None of these.

isi-entrance 2011 Q10 Counting Functions with Constraints View

Let $A$ be the set $\{ 1,2 , \ldots , 6 \}$. How many functions from $A$ to $A$ are there such that the range of $f$ has exactly 5 elements?
(a) 240
(b) 720
(c) 1800
(d) 10800

isi-entrance 2012 Q6 Permutation Properties and Enumeration (Abstract) View

Let $f:\{1,2,3,4\} \to \{1,2,3,4\}$ be a function such that $f(i) \neq i$ for all $i$ (i.e., a derangement). Find the number of such functions.

isi-entrance 2013 Q2 4 marks Forming Numbers with Digit Constraints View

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

isi-entrance 2015 QB1 Forming Numbers with Digit Constraints View

Find the sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4,5$, each digit appearing at most once.

isi-entrance 2016 Q2 4 marks Forming Numbers with Digit Constraints View

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

isi-entrance 2016 Q2 4 marks Forming Numbers with Digit Constraints View

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

isi-entrance 2017 Q7 Permutation Properties and Enumeration (Abstract) View

Let $A = \{ 1,2 , \ldots , n \}$. For a permutation $P = ( P ( 1 ) , P ( 2 ) , \cdots , P ( n ) )$ of the elements of $A$, let $P ( 1 )$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i , j \in A$:

if $i < j < P ( 1 )$, then $j$ appears before $i$ in $P$; and
if $P ( 1 ) < i < j$, then $i$ appears before $j$ in $P$.

isi-entrance 2017 Q14 Counting Functions or Mappings with Constraints View

Let $A = \{1,2,3,4,5,6\}$ and $B = \{a,b,c,d,e\}$. How many functions $f : A \rightarrow B$ are there such that for every $x \in A$, there is one and exactly one $y \in A$ with $y \neq x$ and $f(x) = f(y)$?
(A) 450
(B) 540
(C) 900
(D) 5400.

isi-entrance 2017 Q23 Linear Arrangement with Constraints View

Consider all the permutations of the twenty six English letters that start with $z$. In how many of these permutations the number of letters between $z$ and $y$ is less than those between $y$ and $x$?
(A) $6 \times 23!$
(B) $6 \times 24!$
(C) $156 \times 23!$
(D) $156 \times 24!$.

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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