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.
\[
\left({ x\above 0.0pt
x - 2}\right)\cdot \left({x\above 0.0pt
2}\right) - 11\cdot \left({x\above 0.0pt
2}\right) + 28 < 0
\]
\(\{4\}\)
\(\{5;6\}\)
\((4;7)\)
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?
\(18\)
\(\frac{18!}
{3!\; 15!}=816\)
\(18\cdot 16\cdot 3=864\)
\(20\cdot 19\cdot 18=6\:840\)
9000139303
Level:
A
A DJ's playlist contains \(18\)
songs. In this list there are \(7\)
rap songs, \(5\)
oldies and \(6\)
rock songs. The opening part should consist of one rap song, two oldies and one rock
song. The order of the songs does not matter. Find the number of possible ways how
to put the opening together.
\(420\)
\(120\)
\(320\)
\(520\)
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?
\(\frac{18!}
{3!\; 15!}\)
\(18\)
\(18\cdot 16\cdot 3\)
\(20\cdot 19\cdot 18\)
9000139304
Level:
A
Find the number of possibilities how to choose a pair of timekeepers for a sport event if
there are \(50\)
candidates available for this job.
\(\frac{50!}
{2!\; 48!} = 1\:225\)
\(\frac{50}
{2} = 25\)
\(50^{2} = 2\:500\)
\(\frac{50!}
{48!} = 2\:450\)
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).
\(\frac{15!}
{9!} =3\:603\:600\)
\(6^{15}=470\:184\:984\:576\)
\(15!\, 6!=941\:525\:544\:960\:000\)
\(\frac{15!}
{9!\, 6!}=5\:005\)