9000140508
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)!}
{n! - (n + 1)!}
\]
\(- 1 - \frac{1}
{n}\)
\(n + 1\)
\(n! + 1\)
\(- \frac{n+1}
{(n-1)!}\)
Combinatorics
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 \(\frac{72!}
{70!+71!}\).
\(71\)
\(72\)
\(\frac{72}
{141}\)
\(\frac{72!}
{141!}\)
9000139301
Level:
A
Five different pieces of fruit are there on the plate. In how many ways it is possible
to distribute them into five children, if each child should have one piece?
\(120\)
\(100\)
\(140\)
\(160\)
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.
\(\frac{50!}
{2!\, 48!} = 1\:225\)
\(\frac{50}
{2} = 25\)
\(50^{2} = 2\:500\)
\(\frac{50!}
{48!} = 2\:450\)
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?
\(120\)
\(320\)
\(520\)
\(720\)
9000139305
Level:
A
There are five rooms with three beds and one room with five beds in a hotel. A group of
\(20\)
people booked rooms at this hotel. Determine the number of possible choices for the
people to the five-bed room.
\(\frac{20!}
{5!\, 15!}=15\:504\)
\(20\cdot 3\cdot 5=300\)
\(\frac{20!}
{3!\, 5!}=3\:379\:030\:566\:912\:000\)
\(20^{5}=3\:200\:000\)