9000140504
Level:
B
Simplify for \(n\in \mathbb{N}\),
\(n\geq 2\).
\[
\frac{n\cdot (n - 2)!}
{(n - 1)\cdot n!}
\]
\(\frac{1}
{(n-1)^{2}} \)
\(\frac{(n^{2}-2n)!}
{(n^{2}-n)!} \)
\(\frac{n+1}
{n-1}\)
\(\frac{(n-2)!}
{(n-1)!}\)
Combinatorics
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?
\(\frac{5!}
{3!\, 2!}=10\)
\(\frac{5!}
{3!}=20\)
\(3^{3}=27\)
\(3!=6\)
9000136903
Level:
B
Simplify \(\left({4\above 0.0pt
0}\right) +\left ({4\above 0.0pt
1}\right) +\left ({4\above 0.0pt
2}\right) +\left ({4\above 0.0pt
3}\right) +\left ({4\above 0.0pt
4}\right)\).
\(4^{2}\)
\(14\)
\(\left({5\above 0.0pt
4}\right)\)
\(32\)
\(\left({8\above 0.0pt
4}\right)\)
9000136904
Level:
B
Simplify \(\left({n+1\above 0.0pt
n} \right) +\left ({n+1\above 0.0pt
1} \right)\)
for \(n\in \mathbb{N}\).
\(2(n + 1)\)
\(n + 2\)
\(\left({n+2\above 0.0pt
n} \right)\)
\(2\)
\(\left (n + 1\right )^{2}\)
9000136902
Level:
B
Simplify \(\left({12\above 0.0pt
10}\right) -\left ({12\above 0.0pt
2} \right)\).
\(0\)
\(66\)
\(\left({12\above 0.0pt
8} \right)\)
\(1\)
\(\left({12\above 0.0pt
0} \right)\)
9000136907
Level:
B
Solve the equation involving binomial coefficients.
\[
\left({x + 1\above 0.0pt
x} \right) -\left ({x + 1\above 0.0pt
x + 1}\right) = 21
\]
\(21\)
\(20\)
\(11\)
\(10\)
The equation has no solution.
9000136906
Level:
B
Solve the equation involving binomial coefficient.
\[
\left({x\above 0.0pt
2}\right) = 15
\]
\(6\)
\(5\)
\(4\)
\(3\)
The equation has no solution.
9000136908
Level:
B
Solve the equation involving binomial coefficients.
\[
\left({x + 1\above 0.0pt
x} \right) -\left ({x + 1\above 0.0pt
1} \right) = 1
\]
The equation has no solution.
\(- 1\)
\(1\)
\(2\)
\(0\)