9000140501
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{n!}
{(n - 1)!}
\]
\(n\)
\(\frac{n}
{n-1}\)
\(\frac{n!}
{n!-1!}\)
\(- 1\)
Combinatorics
9000139707
Level:
A
A Morse code utilized dots and dashes to encode letters of
an alphabet. Find the number of signals of the length from
\(1\) to
\(4\) which
can be obtained from dots and dashes.
\(2 + 2^{2} + 2^{3} + 2^{4}=30\)
\(1 + 2 + 3! + 4!=33\)
\(\frac{4!}
{3!\, 2!}=2\)
\(2 \cdot 1 + 2 \cdot 2 + 2 \cdot 3 + 2 \cdot 4=20\)
9000140504
Level:
B
Simplify for \(n\in \mathbb{N}\),
\(n\geq 2\).
\[
\frac{n\cdot (n - 2)!}
{(n - 1)\cdot n!}
\]
\(\frac{1}
{(n-1)^{2}} \)
\(\frac{(n^{2}-2n)!}
{(n^{2}-n)!} \)
\(\frac{n+1}
{n-1}\)
\(\frac{(n-2)!}
{(n-1)!}\)
9000139708
Level:
A
The shelf contains \(15\)
books. From this amount, \(9\)
books are in English and \(6\)
books in other languages. Find the number of possibilities how to rearrange the
books on the shelf, if all English books have to be on the left and the other on the
right.
\(9!\, 6!=261\:273\:600\)
\(9^{6}=531\:441\)
\(\frac{9!}
{6!}=504\)
\(\frac{9!}
{6!\, 3!}=84\)
9000140506
Level:
B
Simplify \(\frac{2!}
{1!} + \frac{3!}
{2!} + \frac{4!}
{3!}\).
\(9\)
\(\frac{29}
{6} \)
\(\frac{9}
{6}\)
\(\frac{12!+9!+8!}
{6!} \)
9000139710
Level:
C
The wallet contains nine coins: three
\(1\)-Euro coins,
three \(2\)-Euro coins
and three \(5\)-Euro
coins. How many different amounts can be paid if we have to pay the amount exactly
and use just three coins for this payment?
\(\frac{5!}
{3!\, 2!}=10\)
\(\frac{5!}
{3!}=20\)
\(3^{3}=27\)
\(3!=6\)
9000140507
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{(n + 2)!}
{n! + (n + 1)!}
\]
\(n + 1\)
\(\frac{n!+2}
{2n!+1}\)
\(\frac{n+2}
{2n+1}\)
\(\frac{n!+2!}
{2n!+1!}\)
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?
\(\frac{11!}
{5!\, 4!\, 2!}=6\:930\)
\(5\cdot 4\cdot 2=40\)
\(5!\, 4!\, 2!=5\:760\)
\(\left (5!\, 4!\right )^{2}=8\:294\:400\)