9000140507
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 2)!}
{n! + (n + 1)!}
\]
\(n + 1\)
\(\frac{n!+2}
{2n!+1}\)
\(\frac{n+2}
{2n+1}\)
\(\frac{n!+2!}
{2n!+1!}\)
Combinatorics
9000139307
Level:
A
There are \(10\)
students hurrying to the dining room. Find the number of possibilities how can be
these students ordered into the row to get the meal.
\(10!=3\:628\:800\)
\(10\)
\(10^{10}\)
\(\frac{10!}
{10} =362\:880\)
9000140508
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)!}
{n! - (n + 1)!}
\]
\(- 1 - \frac{1}
{n}\)
\(n + 1\)
\(n! + 1\)
\(- \frac{n+1}
{(n-1)!}\)
9000139703
Level:
A
The box contains \(5\)
red crayons, \(4\)
yellow crayons and \(2\)
green crayons. The crayons are removed from the box and arranged in a
line. How many different color patterns can be obtained by this procedure?
\(\frac{11!}
{5!\, 4!\, 2!}=6\:930\)
\(5\cdot 4\cdot 2=40\)
\(5!\, 4!\, 2!=5\:760\)
\(\left (5!\, 4!\right )^{2}=8\:294\:400\)
9000140502
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 1)!}
{(n - 1)!}
\]
\(n^{2} + n\)
\((n + 1)^{2}\)
\(\frac{n+1}
{n-1}\)
\(- 1\)
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?
\(36^{4}=1\:679\:616\)
\(10\cdot 26^{4}=4\:569\:760\)
\(\frac{36!}
{32!\, 4!}=58\:905\)
\(\frac{26!}
{22!\, 4!}=14\:950\)