9000141506
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\left({11\above 0.0pt
4} \right) =\left ({11\above 0.0pt
x} \right)
\]
\(\{4,7\}\)
\(\{4\}\)
\(\{ - 4\}\)
Combinatorics
9000141507
Level:
B
Assuming \(x,y\in \mathbb{N}\),
find the solution set of the following equation.
\[
\left({x\above 0.0pt
y}\right)^{2} - 2\cdot \left({x\above 0.0pt
y}\right) - 3 = 0
\]
\(\{[3,1],[3,2]\}\)
\(\{[3,1]\}\)
\(\{[3,1],[1,3]\}\)
9000141509
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following inequality.
\[
2\cdot \left({x - 1\above 0.0pt
x - 3}\right) + x\cdot (x - 9)\leq - 8
\]
\(\{3,4,5\}\)
\(\{1,2,3,4,5\}\)
\([ 1,5] \)
9000141510
Level:
B
Assuming \(x\in \mathbb{N}\),
\(n\geq 2\), find the solution set of the following inequality.
\[
\left({ x\above 0.0pt
x - 2}\right)\cdot \left({x\above 0.0pt
2}\right) - 11\cdot \left({x\above 0.0pt
2}\right) + 28 < 0
\]
\(\{4\}\)
\(\{5,6\}\)
\((4,7)\)
9000139308
Level:
A
The shooting club has \(25\)
members. Among the members it is necessary to vote a board: a president, a cashier
and a webmaster. One person cannot have more than one of these positions and there
is only one member skilled enough to be a webmaster. How many possibilities exist to
set up the board?
\(24\cdot 23=552\)
\(25\cdot 24=600\)
\(24\cdot 23\cdot 22=12\:144\)
\(25\cdot 24\cdot 23=13\:800\)
9000139306
Level:
A
How many different ice cream bowls can be prepared from
\(8\) kinds
of an ice cream if the bowl contains just three different kinds of ice cream?
\(\frac{8!}
{3!\, 5!}=56\)
\(\frac{8!}
{2!\, 5!}=168\)
\(\frac{8!}
{5!}=336\)
\(8!=40\:320\)
9000140510
Level:
B
Find the solution set of the equation assuming
\(x\in \mathbb{N}\).
\[
(x + 1)! + x! = 6x + 12
\]
\(\{3\}\)
\(\{2\}\)
\(\{1\}\)
\(\{4\}\)
9000139309
Level:
A
There are \(20\) tablets in an
e-shop. From this amount \(18\)
tablets are new and \(2\)
tablets have been returned by customers. The e-shop manager gets an order
containing three tablets and he wants to get rid of the returned tablets first. How
many possibilities exist to complete the order?
\(18\)
\(\frac{18!}
{3!\, 15!}=816\)
\(18\cdot 16\cdot 3=864\)
\(20\cdot 19\cdot 18=6\:840\)
9000139701
Level:
A
There are \(15\)
athletes in an athletic meeting. Determine in how many ways it is possible to
obtain the results on the first six places of the scoreboard if the place on
scoreboard cannot be shared (one athlete per one place on scoreboard).
\(\frac{15!}
{9!} =3\:603\:600\)
\(6^{15}=470\:184\:984\:576\)
\(15!\, 6!=941\:525\:544\:960\:000\)
\(\frac{15!}
{9!\, 6!}=5\:005\)
9000139310
Level:
A
There are \(20\) tablets in an
e-shop. From this amount \(18\)
tablets are new and \(2\)
tablets have been returned by customers. The e-shop manager gets an order
containing three tablets and he wants to use only the new tablets for this order. How
many possibilities exist to complete the order?
\(\frac{18!}
{3!\, 15!}\)
\(18\)
\(18\cdot 16\cdot 3\)
\(20\cdot 19\cdot 18\)