9000140502
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)!}
{(n - 1)!}
\]
\(n^{2} + n\)
\((n + 1)^{2}\)
\(\frac{n+1}
{n-1}\)
\(- 1\)
Combinatorics
9000139704
Level:
C
There are \(5\)
different kinds of cakes in a shop. Find the number of possibilities how to buy
\(8\) cakes in this shop.
(There is more than \(8\)
cakes of each kind available.)
\(\frac{12!}
{8!\, 4!}=495\)
\(5!\, 8!=4\:838\:400\)
\(5^{8}=390\:625\)
\(\frac{8!}
{5!\, 3!}=56\)
9000140503
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)! + (n - 1)!}
{n!}
\]
\(\frac{n^{2}+n+1}
{n} \)
\(2\)
\(n^{2} - 1\)
\(\frac{n^{2}-n+1}
{n} \)
9000139705
Level:
A
From the group of \(10\) boys
and \(5\) girls we have to
select a small group of \(3\)
boys and \(2\)
girls. How many possibilities exist for this choice?
\(\frac{10!}
{7!\, 3!}\cdot \frac{5!}
{3!\, 2!}=1\:200\)
\(5^{10}=9\:765\:625\)
\(10\cdot 5!\, 3!=7\:200\)
\(5\cdot \frac{10!}
{3!} =3\:024\:000\)
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?
\(120\)
\(100\)
\(140\)
\(160\)
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?
\(36^{4}=1\:679\:616\)
\(10\cdot 26^{4}=4\:569\:760\)
\(\frac{36!}
{32!\, 4!}=58\:905\)
\(\frac{26!}
{22!\, 4!}=14\:950\)
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 \(\frac{72!}
{70!+71!}\).
\(71\)
\(72\)
\(\frac{72}
{141}\)
\(\frac{72!}
{141!}\)