9000141503
Level:
B
Let \(A\) be a
set with \(n\)
mutually different elements. If two elements are removed from
\(A\), the number of all
permutations of set \(A\)
decreases \(20\)-times.
Find \(n\).
\(5\)
\(4\)
\(5\) or
\(- 4\)
Combinatorics
9000141504
Level:
B
Let \(A\) be a
set with \(n\)
mutually different elements. If we add one element to the set
\(A\), the number of
\(3\)-combinations
of a set \(A\) is
increased by \(21\).
Find \(n\).
\(7\)
\(6\)
\(43\)
9000141501
Level:
B
Let \(A\) be set with
\(n\) mutually different
elements. If \(n\) is increased
by \(2\), then number
of \(3\)-permutations
is increased by \(384\).
Find \(n\). (The term
„\(k\)-permutation” stands for
an ordered arrangement of \(k\)
objects from a set of \(n\)
objects.)
\(8\)
\(64\)
\(32\)
9000141510
Level:
B
Assuming \(x\in \mathbb{N}\),
\(n\geq 2\), find the solution set of the following inequality.
\[
\left({ x\above 0.0pt
x - 2}\right)\cdot \left({x\above 0.0pt
2}\right) - 11\cdot \left({x\above 0.0pt
2}\right) + 28 < 0
\]
\(\{4\}\)
\(\{5;6\}\)
\((4;7)\)
9000139302
Level:
A
The phone number contains nine digits. A witness does not remember
the full number, but he remembers that the phone number starts by
\(728\), ends
by \(01\)
and there is no repeating digit in the number. How many phone numbers meet these
conditions?
\(120\)
\(320\)
\(520\)
\(720\)
9000139709
Level:
A
Find the number of possible ways how to organize a group of
\(10\)
children in one ordered row if Mike and Paul should be next to each other.
\(2\cdot 9!=725\:760\)
\(10!=3\:628\:800\)
\(\frac{10!}
{2!} =1\:814\:400\)
\(\frac{10!}
{8!} =90\)
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?
\(24\cdot 23=552\)
\(25\cdot 24=600\)
\(24\cdot 23\cdot 22=12\:144\)
\(25\cdot 24\cdot 23=13\:800\)
9000140505
Level:
B
Simplify \(\frac{72!}
{70!+71!}\).
\(71\)
\(72\)
\(\frac{72}
{141}\)
\(\frac{72!}
{141!}\)
9000140509
Level:
B
Find the solution set of the equation assuming
\(x\in \mathbb{N}\).
\[
(x + 1)! = 6(x - 1)!
\]
\(\{2\}\)
\(\{ - 3;\ 2\}\)
\(\left \{\frac{5}
{7}\right \}\)
\(\left \{\frac{7}
{5}\right \}\)
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?
\(\frac{8!}
{3!\; 5!}=56\)
\(\frac{8!}
{2!\; 5!}=168\)
\(\frac{8!}
{5!}=336\)
\(8!=40\:320\)