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5 votes
3 answers
123 views

n students standing in line are to be divided into teams and in each team appointed a captain. How many ways to do that? (I need last transformation)

5 votes
0 answers
57 views

In how many ways can numbers $ \in \{1, 2, ..., 3n \}$ be arranged in such way that the sum of every $3$ consecutive numbers is divisible by $3$?

4 votes
0 answers
102 views

If a circle can be inscribed in two quadrilaterals, then circle can be inscribed also in the quadrilateral $ABCD$

4 votes
1 answer
73 views

Evaluate $\int_{\gamma} \frac{2 \sin(z) +e^z}{z^2 - 2z} dz$ where $\gamma = C(0,3)$ with positive orientation.

4 votes
0 answers
72 views

What's the probability that the only coin left is fair given that exactly $n$ tosses were required to obtain heads with each of the first two coins?

4 votes
3 answers
129 views

Combinatorial proof $\sum^n_{k=0} {{2n}\choose{k}}{{n}\choose{k}}{{2n-k}\choose{n}} = {{2n}\choose{n}}^2$

4 votes
1 answer
73 views

Different sidewalks that can be laid using n $1 \times 1$ square white, blue and red tiles so that no two red tiles lie next to each other?

3 votes
3 answers
186 views

Determine the image of the unit circle $S^1$ by the action of the matrix $e^A$.

3 votes
2 answers
398 views

A point is randomly selected from interior of a square with geom. prob. Find the distrib. of the area of a rectangle, its expected value and variance.

3 votes
0 answers
60 views

Prove combinatoric equation: $\sum_{k=1}^n{{k}\choose{j}}k = {{n+1}\choose{j+1}}n - {{n+1}\choose{j+2}}$

3 votes
1 answer
66 views

There are $n$ balls, independence of events $A_j =$ {number j appeared in first j draws}, where: $j = 1, 2, . . . , n$. Need to verify a solution.

3 votes
2 answers
76 views

We take k-element sequences with values from the set $[n]$. For each, we find the smallest value, and then sum these. Why the sum $=1^k+...+ n^k$?

2 votes
2 answers
154 views

Heptagon is divided into pentagons and hexagons. Prove that there are at least $27$ pentagons in this division.

2 votes
1 answer
106 views

How many paths of length $n$ with given beginning and end are there in the graph?

2 votes
1 answer
85 views

Number of sequences $A_1, A_2, ..., A_k$ such that: $A_1 \subseteq A_2 \subseteq ... \subseteq A_k = [n]$.

2 votes
2 answers
101 views

The probability that the first ball drawn was white, if it is known that after two draws urn II contains exactly two white balls.

2 votes
1 answer
50 views

Probability that as many heads came out as tails, with the $12$th toss being the first toss after which the numbers of heads and tails became equal.

2 votes
2 answers
94 views

Determine the number of $5$ digit numbers in which no digit occurs more than twice (numbers starting with $0$ are allowed).

2 votes
0 answers
78 views

There two types of tiles, we need to construct a $2 \times n$ rectangle filled with them. How many ways are there to do that?

2 votes
0 answers
55 views

Sequences of length $n$ made of digits $2, 3, 5, 6$ that no element of sequence divides the next one and that the first element is $5$.

2 votes
1 answer
62 views

A table is made of $\frac{n(n + 1)}{2}$ boxes where we randomly write numbers. Calculate the probability that $\max_1 < \max_2 < ... < \max_n$

2 votes
1 answer
43 views

Number of ways to select a subset of the set $ \{1, 2, . . . , 200 \}$ in such a way that it contains the same number of even and odd elements.

2 votes
1 answer
91 views

Show that the measure $\mu ([a,b]) = \ln \left( \frac{1+b}{1+a} \right)$ is invariant for function $\Phi_k$.

2 votes
1 answer
61 views

$A$ is a borel subset of $\mathbb{R}^2$ and we know that $\lambda_2(A) = \pi$. Prove that: $\int_A (x^2 + y^2) \ d \lambda_2(x,y) \geq \frac{\pi}{2}$

2 votes
1 answer
54 views

We know that $g \in L_1$ and that $g*g \leq g$ nearly everywhere. Show that: $\int_{\mathbb{R}} g(x) d\lambda_1(x) \leq 1$

2 votes
1 answer
106 views

Gambler's ruin with asymmetric probabilities

2 votes
1 answer
138 views

$M \subset \mathbb{R}^3$ is a rotation of set $\{ x^2 + z^2 = 4x-3, 1<x<2, -1<z<1 \}$ by $\pi$ around axis OZ. Find $\int_M\max(x,y,z) d \lambda_2$

2 votes
2 answers
125 views

$f:[0,1] \to R$ is measurable but not Lebesgue integrable. Is $g(x,y) := f(x) - f(y)$ is also not Lebesgue integrable on $[0, 1] × [0, 1]$?

2 votes
1 answer
36 views

Average number of calls coming into office X, in one hour, is $4$, to Y is $7$. What is the prob. that on some day $100$ calls will come to both?

2 votes
0 answers
32 views

$24$ meteorites reach the Earth's surface each year. Prob. that no more than $3$ such meteorites will come in one month. Solution verification.

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