LFM Stats And Pure

csat-suneung 2020 Q19 4 marks Word Permutations with Repeated Letters View

From the numbers $1,2,3,4,5,6$, five numbers are selected with repetition allowed to satisfy the following conditions, and then all five-digit natural numbers that can be formed by arranging them in a line are counted. What is the number of such natural numbers? [4 points] (가) Each odd number is either not selected or selected exactly once. (나) Each even number is either not selected or selected exactly twice.
(1) 450
(2) 445
(3) 440
(4) 435
(5) 430

csat-suneung 2020 Q22 3 marks Factorial and Combinatorial Expression Simplification View

Find the value of ${ } _ { 7 } \mathrm { P } _ { 2 } + { } _ { 7 } \mathrm { C } _ { 2 }$. [3 points]

csat-suneung 2020 Q28 4 marks Forming Numbers with Digit Constraints View

From the numbers $1,2,3,4,5,6$, select five numbers with repetition allowed and arrange them in a line to form a five-digit natural number, satisfying the following conditions. How many such five-digit natural numbers can be formed? [4 points] (가) Each odd number is either not selected or selected exactly once. (나) Each even number is either not selected or selected exactly twice.

csat-suneung 2021 Q15 4 marks Circular Arrangement View

There are 6 students including students $\mathrm { A }$, $\mathrm { B }$, and $\mathrm { C }$. These 6 students sit around a circular table at equal intervals satisfying the following conditions. How many ways are there to seat them? (Note: arrangements that coincide by rotation are considered the same.) [4 points] (가) A and B are adjacent. (나) B and C are not adjacent.
(1) 32
(2) 34
(3) 36
(4) 38
(5) 40

csat-suneung 2021 Q26 4 marks Circular Arrangement View

There are 6 students including three students $\mathrm { A } , \mathrm { B } , \mathrm { C }$.
Find the number of ways these 6 students can all sit around a circular table with equal spacing satisfying the following conditions. (Note: rotations that coincide are considered the same.) [4 points] (가) A and B are adjacent. (나) B and C are not adjacent.

csat-suneung 2023 Q24 3 marks Forming Numbers with Digit Constraints View

Among four-digit natural numbers that can be formed by selecting 4 numbers from the digits 1, 2, 3, 4, 5 with repetition allowed and arranging them in a line, how many are odd numbers greater than or equal to 4000? [3 points]
(1) 125
(2) 150
(3) 175
(4) 200
(5) 225

csat-suneung 2023 Q30 4 marks Counting Functions with Constraints View

For the set $X = \{ x \mid x \text{ is a natural number not exceeding } 10 \}$, find the number of functions $f : X \rightarrow X$ satisfying the following conditions. [4 points] (가) For all natural numbers $x$ not exceeding 9, $f ( x ) \leq f ( x + 1 )$. (나) When $1 \leq x \leq 5$, $f ( x ) \leq x$, and when $6 \leq x \leq 10$, $f ( x ) \geq x$. (다) $f ( 6 ) = f ( 5 ) + 6$

csat-suneung 2024 Q23 2 marks Word Permutations with Repeated Letters View

The number of ways to arrange all 5 letters $x, x, y, y, z$ in a row is? [2 points]
(1) 10
(2) 20
(3) 30
(4) 40
(5) 50

csat-suneung 2024 Q25 3 marks Probability via Permutation Counting View

There are 6 cards with the numbers $1, 2, 3, 4, 5, 6$ written on them, one number per card. When all 6 cards are arranged in a row in random order using each card exactly once, find the probability that the sum of the two numbers on the cards at both ends is at most 10. [3 points]
(1) $\frac{8}{15}$
(2) $\frac{19}{30}$
(3) $\frac{11}{15}$
(4) $\frac{5}{6}$
(5) $\frac{14}{15}$

csat-suneung 2025 Q28 4 marks Counting Functions with Constraints View

For the set $X = \{1, 2, 3, 4, 5, 6\}$, how many functions $f : X \rightarrow X$ satisfy the following conditions? [4 points] (가) The value of $f(1) \times f(6)$ is a divisor of 6. (나) $2f(1) \leq f(2) \leq f(3) \leq f(4) \leq f(5) \leq 2f(6)$
(1) 166
(2) 171
(3) 176
(4) 181
(5) 186

csat-suneung 2026 Q23 2 marks Linear Arrangement with Constraints View

How many ways are there to select 3 letters from the four letters $a , b , c , d$ with repetition allowed and arrange them in a row? [2 points]
(1) 56
(2) 60
(3) 64
(4) 68
(5) 72

csat-suneung 2026 Q30 4 marks Distribution of Objects into Bins/Groups View

There are 10 empty bags arranged in a row, and 8 balls. Distribute the balls into the bags so that each bag contains at most 2 balls. Find the number of cases satisfying the following conditions. (Here, the balls are indistinguishable from each other.) [4 points] (가) The number of bags containing 1 ball is either 4 or 6. (나) Bags adjacent to a bag containing 2 balls contain no balls.

gaokao 2015 Q2 Selection and Task Assignment View

2. In a marathon competition, the stem-and-leaf plot in Figure 1 shows the results (in minutes) of 35 athletes. [Figure]
If the athletes are numbered 1-35 according to their results from best to worst, and 7 people are selected using systematic sampling, then the number of athletes with results in the interval [139, 151] is
A. 3
B. 4
C. 5
D. 6

gaokao 2015 Q6 Forming Numbers with Digit Constraints View

6. Using the digits $0, 1, 2, 3, 4, 5$ to form five-digit numbers with no repeated digits, the number of even numbers greater than $40000$ is
(A) $144$
(B) $120$
(C) $96$
(D) $72$

gaokao 2015 Q15 13 marks Probability via Permutation Counting View

15. (13 points) Three table tennis associations have 27, 9, and 18 members respectively. Using stratified sampling, 6 athletes are selected from these three associations to participate in a competition. (I) Find the number of athletes to be selected from each of the three associations respectively; (II) The 6 selected athletes are numbered $A _ { 1 } , A _ { 2 } , A _ { 3 } , A _ { 4 } , A _ { 5 } , A _ { 6 }$ respectively. Two athletes are randomly selected from these 6 athletes to participate in a doubles match.
(i) List all possible outcomes using the given numbering;
(ii) Let event $A$ be ``at least one of the two athletes numbered $A _ { 5 }$ and $A _ { 6 }$ is selected''. Find the probability of event $A$ occurring.

gaokao 2017 Q6 Linear Arrangement with Constraints View

6. The number of ways to arrange 3 people in a row is
A. $12$ ways
B. $18$ ways
C. $24$ ways
D. $36$ ways [Figure] [Figure] [Figure] [Figure] [Figure]
C. C and D can know each other's scores
B. B can know all four people's scores

gaokao 2019 Q10 Linear Arrangement with Constraints View

10. Given that the foci of ellipse $C$ are $F _ { 1 } ( - 1,0 ) , F _ { 2 } ( 1,0 )$, and a line through $F _ { 2 }$ intersects $C$ at points $A , B$. If $\left| A F _ { 2 } \right| = 2 \left| F _ { 2 } B \right| , | A B | = \left| B F _ { 1 } \right|$, then the equation of $C$ is
A. $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$
B. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$
C. $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$
D. $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$

gaokao 2021 Q10 Probability via Permutation Counting View

10. Three 1's and two 0's are randomly arranged in a row. The probability that the two 0's are not adjacent is
A. 0.3
B. 0.5
C. 0.6
D. 0.8

gaokao 2021 Q10 Probability via Permutation Counting View

10. Arrange 4 ones and 2 zeros randomly in a row. The probability that the 2 zeros are not adjacent is [Figure]
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{2}{3}$
D. $\frac{4}{5}$

grandes-ecoles 2016 QI.A.1 Combinatorial Structures on Permutation Matrices/Groups View

Justify that $\mathcal{X}_n$ is a finite set and determine its cardinality.

grandes-ecoles 2016 QII.A.1 Counting Functions with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
What is $S(p, n)$ for $p < n$?

grandes-ecoles 2016 QII.A.2 Counting Functions with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
Determine $S(n, n)$.

grandes-ecoles 2017 QIA Combinatorial Proof or Identity Derivation View

Let $k$ and $n$ be two strictly positive integers. Show that there exists only a finite number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts.

grandes-ecoles 2017 QIB Combinatorial Proof or Identity Derivation View

Throughout the problem, for every pair $( n , k )$ of strictly positive integers, we denote by $S ( n , k )$ the number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts. We further set $S ( 0,0 ) = 1$ and, for all $( n , k ) \in \mathbb { N } ^ { * 2 } , S ( n , 0 ) = S ( 0 , k ) = 0$.
Express $S ( n , k )$ as a function of $n$ or of $k$ in the following cases:
I.B.1) $k > n$;
I.B.2) $k = 1$.

grandes-ecoles 2017 QIC Combinatorial Proof or Identity Derivation View

Throughout the problem, for every pair $( n , k )$ of strictly positive integers, we denote by $S ( n , k )$ the number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts. We further set $S ( 0,0 ) = 1$ and, for all $( n , k ) \in \mathbb { N } ^ { * 2 } , S ( n , 0 ) = S ( 0 , k ) = 0$.
Show that for all strictly positive integers $k$ and $n$, we have $$S ( n , k ) = S ( n - 1 , k - 1 ) + k S ( n - 1 , k )$$ One may distinguish the partitions of $\llbracket 1 , n \rrbracket$ according to whether or not they contain the singleton $\{ n \}$.

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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