LFM Stats And Pure

germany-abitur 2023 QA b 3 marks Probability via Permutation Counting View

Give a term with which the probability can be calculated that each number is obtained at least once.

germany-abitur 2024 QB 1e 2 marks Probability via Permutation Counting View

Determine the probability that the first two packages are returns.

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 2016 QII.A.3 Counting Functions or Mappings 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 $S(n+1, n)$.

grandes-ecoles 2016 QII.B.1 Counting Functions or Mappings 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 $S(p, 0) = 0$.
How many applications are there from $\llbracket 1, p \rrbracket$ to $\llbracket 1, n \rrbracket$?

grandes-ecoles 2016 QII.B.2 Counting Functions or Mappings 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 $S(p, 0) = 0$.
For $p \geqslant n$, establish the formula $$n^p = \sum_{k=0}^{n} \binom{n}{k} S(p, k)$$ where $S(p, 0) = 0$ by convention.

grandes-ecoles 2016 QII.B.3 Counting Functions or Mappings 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 $S(p, 0) = 0$.
Deduce an expression of $S(p, n)$ for $p \geqslant n$.

grandes-ecoles 2016 QII.B.4 Counting Functions or Mappings 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 $S(p, 0) = 0$.
By rereading question I.B.5, comment on the consistency of the expression of $S(p,n)$ for $p < n$.

grandes-ecoles 2019 Q14 Combinatorial Counting (Non-Probability) View

We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, an outcome resulting from $n$ successive draws is modeled by the $n$-tuple indicating the color and number of the balls successively obtained. We denote by $\Omega_{n}$ the set of possible outcomes of these $n$ draws.
By examining the number of balls in the urn just before each draw, justify that, for $n \geqslant 1$, $$\operatorname{card}(\Omega_{n}) = (a_{0} + b_{0}) \times \cdots \times (a_{0} + b_{0} + s(n-1)) = s^{n} L_{n}\left(\frac{a_{0} + b_{0}}{s}\right).$$

grandes-ecoles 2019 Q32 Permutation Properties and Enumeration (Abstract) View

A permutation $\sigma$ of $\llbracket 1, n \rrbracket$ is called alternating up if the list $(\sigma(1), \ldots, \sigma(n))$ satisfies $x_1 < x_2 > x_3 < x_4 > \cdots$. Determine the alternating up permutations of $\llbracket 1, n \rrbracket$ for $n = 2$, $n = 3$, $n = 4$.

grandes-ecoles 2019 Q33 Permutation Properties and Enumeration (Abstract) View

A permutation $\sigma$ of $\llbracket 1, n \rrbracket$ is called alternating up if $(\sigma(1), \ldots, \sigma(n))$ satisfies $x_1 < x_2 > x_3 < x_4 > \cdots$, and alternating down if it satisfies the reverse inequalities. Show, for every $n \geqslant 2$, that the number of alternating up permutations equals the number of alternating down permutations.

grandes-ecoles 2019 Q34 Permutation Properties and Enumeration (Abstract) View

Let $\beta_n$ denote the number of alternating up permutations of $\llbracket 1, n \rrbracket$ (with $\beta_0 = \beta_1 = 1$). Let $k$ and $n$ be two integers such that $2 \leqslant k \leqslant n$ and $A$ a subset with $k$ elements of $\llbracket 1, n \rrbracket$. We consider the lists $(x_1, \ldots, x_k)$ consisting of $k$ pairwise distinct elements of $A$. Show that the number of these lists that are alternating up equals $\beta_k$.

grandes-ecoles 2019 Q35 Combinatorial Proof or Identity Derivation View

Let $\beta_n$ denote the number of alternating up permutations of $\llbracket 1, n \rrbracket$ (with $\beta_0 = \beta_1 = 1$). For $k \in \llbracket 0, n \rrbracket$, enumerate the permutations $\sigma$ alternating (up or down) of $\llbracket 1, n+1 \rrbracket$ such that $\sigma(k+1) = n+1$. Show, for every integer $n \geqslant 1$, $$2\beta_{n+1} = \sum_{k=0}^{n} \binom{n}{k} \beta_k \beta_{n-k}.$$

grandes-ecoles 2019 Q37 Probability via Permutation Counting View

For every integer $n \geqslant 2$, let $p_n$ be the probability that a uniformly random permutation of $\llbracket 1, n \rrbracket$ is alternating up (with $p_0 = p_1 = 1$). Show that the sequence $(p_n)$ tends to 0. Give an equivalent of $p_{2n+1}$ as $n$ tends to infinity.

grandes-ecoles 2019 Q38 Deriving or Identifying a Probability Distribution from a Random Process View

For every integer $n \geqslant 2$, equip the set $\Omega_n$ of permutations of $\llbracket 1, n \rrbracket$ with the uniform probability. Let $p_i$ denote the probability that a permutation is alternating up (with $p_0 = p_1 = 1$). Define the random variable $M_n$ on $\Omega_n$ by: $M_n(\sigma) = k+1$ where $k$ is the largest integer such that $(\sigma(1), \ldots, \sigma(k))$ is alternating up. For every $i \in \llbracket 0, n \rrbracket$, show $\mathbb{P}(M_n > i) = p_i$.

grandes-ecoles 2019 Q39 Limit and Convergence of Probabilistic Quantities View

For every integer $n \geqslant 2$, equip the set $\Omega_n$ of permutations of $\llbracket 1, n \rrbracket$ with the uniform probability. Let $p_i$ denote the probability that a permutation is alternating up (with $p_0 = p_1 = 1$). Define the random variable $M_n$ on $\Omega_n$ by: $M_n(\sigma) = k+1$ where $k$ is the largest integer such that $(\sigma(1), \ldots, \sigma(k))$ is alternating up. Express $\mathbb{E}(M_n)$ as a function of $p_0, p_1, \ldots, p_n$. Deduce $\lim_{n \rightarrow \infty} \mathbb{E}(M_n) = \frac{\sin(1) + 1}{\cos(1)}$.

grandes-ecoles 2019 Q44 Evaluation of a Finite or Infinite Sum View

For $j \in \mathbb{N}$, we denote by $Y_{n,j}$ the set of partitions whose first term $\alpha_1$ is less than or equal to $j$ and by $y_{n,j}$ the cardinality of $Y_{n,j}$; we set $y_{0,0} = 1$.
Calculate $y_{n,1}$.

grandes-ecoles 2019 Q45 Functional Equations and Identities via Series View

For $j \in \mathbb{N}$, we denote by $Y_{n,j}$ the set of partitions whose first term $\alpha_1$ is less than or equal to $j$ and by $y_{n,j}$ the cardinality of $Y_{n,j}$; we set $y_{0,0} = 1$.
We propose to show that, if $2 \leqslant j \leqslant n$, then $y_{n,j} = y_{n,j-1} + y_{n-j,\min(j,n-j)}$.
Prove that this equality is true for $j = n$.

grandes-ecoles 2019 Q47 Evaluation of a Finite or Infinite Sum View

For $j \in \mathbb{N}$, we denote by $Y_{n,j}$ the set of partitions whose first term $\alpha_1$ is less than or equal to $j$ and by $y_{n,j}$ the cardinality of $Y_{n,j}$; we set $y_{0,0} = 1$.
Calculate the $y_{n,j}$ for $1 \leqslant j \leqslant n \leqslant 5$ by presenting the results in the form of a table.

grandes-ecoles 2021 Q3 Evaluation of a Finite or Infinite Sum View

Give without proof the value of $C _ { 3 }$ and represent all Dyck paths of length 6.

grandes-ecoles 2021 Q4 Combinatorial Number Theory and Counting View

Let $n \in \mathbb { N }$ and $\gamma = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 n + 2 } \right)$ be a Dyck path of length $2 n + 2$. Let $r = \max \left\{ i \in \llbracket 0 , n \rrbracket \mid s _ { \gamma } ( 2 i ) = 0 \right\}$. We assume $0 < r < n$ and we consider the paths $\alpha = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 r } \right)$ and $\beta = \left( \gamma _ { 2 r + 2 } , \ldots , \gamma _ { 2 n + 1 } \right)$. Justify using a figure that $\gamma _ { 2 r + 1 } = 1 , \gamma _ { 2 n + 2 } = - 1$ and that $\alpha$ and $\beta$ are Dyck paths.

grandes-ecoles 2021 Q22 Combinatorial Identity or Bijection Proof View

We admit that Vandermonde's identity remains valid for all natural integers $u, v, N$: $$\binom{u+v}{N} = \sum_{k=0}^{N} \binom{u}{k} \binom{v}{N-k}.$$ Give a combinatorial interpretation of Vandermonde's identity.

grandes-ecoles 2021 Q28 Combinatorial Proof or Identity Derivation View

Assume that $k$ is even and that $\vec{\imath} \in \mathcal{B}_{k}$ is a cycle passing through $\frac{k}{2} + 1$ distinct vertices (i.e. $|\vec{\imath}| = \frac{k}{2} + 1$). We traverse the edges of $\vec{\imath}$ in order. To each edge of $\vec{\imath}$ we associate an opening parenthesis if this edge appears for the first time and a closing parenthesis if it appears for the second time.
Count the cycles $\vec{\imath}$ that correspond to a fixed well-parenthesized word.

grandes-ecoles 2022 Q4 Recurrence Relations and Sequence Properties View

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality.
Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is finite for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing, and that it is constant from rank $\max ( n , 1 )$ onward.

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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