2010007606
Level:
A
In how many ways can a four-person team be selected from a group of \(10\) students?
\(\frac{10!}
{4!\; 6!}\)
\(\frac{10!}
{4!}\)
\(\frac{10!}
{ 6!}\)
\(10!\)
Combinatorics
2010007605
Level:
B
For \(n\in \mathbb{N}\), the difference \(\left({n+1\above 0.0pt
n} \right) -\left ({ n+1\above 0.0pt
n+1}\right)\)
equals to:
\(n\)
\(0\)
\(n+1\)
\(2(n+1)\)
2010007604
Level:
B
The sum \(\left({19\above 0.0pt
6} \right) +\left ({19\above 0.0pt
7} \right)\) equals to:
\(\left({20\above 0.0pt
7} \right)\)
\(\left({20\above 0.0pt
6} \right)\)
\(\left({19\above 0.0pt
8} \right)\)
\(\left({38\above 0.0pt
13} \right)\)
2010007603
Level:
C
Find the constant term in the expansion of \( \left(2x^2-3\right)^{25} \).
\( -3^{25} \)
\( 3^{25} \)
\( 2^{25} \)
\( -2^{25} \)
2010007602
Level:
C
What is the coefficient at \( x^5 \) in the expansion of \( (1-2x)^{11} \)?
\( -14\:784 \)
\( 14\:784 \)
\( 7\:374 \)
\( -7\:374 \)
2010007601
Level:
A
How many ways are there to arrange the letters in the Czech word LOKOMOTIVA?
\( \frac{10!}{3!} \)
\( \frac{10!}{3} \)
\( \frac{10!}{2!} \)
\( 10!\)
2010007106
Level:
A
Determine the number of four-digit positive integers that can be formed using the digits \(0\), \(1\), \(2\), \(3\), \(4\). The digits can be used repeatedly.
\( 500 \)
\( 96 \)
\( 625 \)
\( 120 \)
2010007105
Level:
A
There are \(20\) girls and \(10\) boys in the class. How many ways are there to designate a president and vice-president of the class if it is required that at least one position will be held by a girl.
\(2\cdot 20\cdot 10 + 20 \cdot 19 =780\)
\(2\cdot 20\cdot 10=400\)
\(20\cdot 19 =380\)
\(20\cdot 10 =200\)
2010007104
Level:
A
There are \(5\) different roads between cities A and B. Find the number of possible ways from the city A to the city B and back, if it is required to use one road from A to B and another different one from B to A.
\( 5 \cdot 4 = 20\)
\( 5 + 4 = 9\)
\( 5 \cdot 5 = 25\)
\( 2 \cdot 5 = 10\)
2010007103
Level:
B
Assuming \(x\in \mathbb{N}\),
\(x \geq 2\), find the solution set of the following inequality.
\[
\left({ x\above 0.0pt
x - 2}\right)\cdot \left({x\above 0.0pt
2}\right) - 20\cdot \left({x\above 0.0pt
2}\right) + 96 < 0
\]
\(\{5\}\)
\(\{9;10;11\}\)
solution does not exist
\( (8;12)\)