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 \(n\).
\(7\)
\(6\)
\(43\)
Combinatorics
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.)
\(8\)
\(64\)
\(32\)
9000141505
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\frac{(x + 2)!}
{x!} = 2\cdot \frac{x!}
{(x - 2)!} + 3!
\]
\(\{4\}\)
\(\{4,1\}\)
\(\{1\}\)
9000141506
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\left({11\above 0.0pt
4} \right) =\left ({11\above 0.0pt
x} \right)
\]
\(\{4,7\}\)
\(\{4\}\)
\(\{ - 4\}\)
9000140508
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)!}
{n! - (n + 1)!}
\]
\(- 1 - \frac{1}
{n}\)
\(n + 1\)
\(n! + 1\)
\(- \frac{n+1}
{(n-1)!}\)
9000139706
Level:
A
The international alphabet contains \(26\)
letters. The letters of this alphabet and the digits from
\(0\) to
\(9\) are used to form a
code of the length \(4\)
(a code contains \(4\)
characters). The characters may repeat through the code and the code is not case
sensitive (uppercase letters are equivalent to lowercase letters). How many codes can
be obtained?
\(36^{4}=1\:679\:616\)
\(10\cdot 26^{4}=4\:569\:760\)
\(\frac{36!}
{32!\, 4!}=58\:905\)
\(\frac{26!}
{22!\, 4!}=14\:950\)
9000140502
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)!}
{(n - 1)!}
\]
\(n^{2} + n\)
\((n + 1)^{2}\)
\(\frac{n+1}
{n-1}\)
\(- 1\)
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.
\(2\cdot 9!=725\:760\)
\(10!=3\:628\:800\)
\(\frac{10!}
{2!} =1\:814\:400\)
\(\frac{10!}
{8!} =90\)
9000140503
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)! + (n - 1)!}
{n!}
\]
\(\frac{n^{2}+n+1}
{n} \)
\(2\)
\(n^{2} - 1\)
\(\frac{n^{2}-n+1}
{n} \)
9000140505
Level:
B
Simplify \(\frac{72!}
{70!+71!}\).
\(71\)
\(72\)
\(\frac{72}
{141}\)
\(\frac{72!}
{141!}\)