9000140501
Level:
B
Simplify for \(n\in \mathbb{N}\).
\[
\frac{n!}
{(n - 1)!}
\]
\(n\)
\(\frac{n}
{n-1}\)
\(\frac{n!}
{n!-1!}\)
\(- 1\)
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.
\(10!=3\:628\:800\)
\(10\)
\(10^{10}\)
\(\frac{10!}
{10} =362\:880\)
9000140504
Level:
B
Simplify for \(n\in \mathbb{N}\),
\(n\geq 2\).
\[
\frac{n\cdot (n - 2)!}
{(n - 1)\cdot n!}
\]
\(\frac{1}
{(n-1)^{2}} \)
\(\frac{(n^{2}-2n)!}
{(n^{2}-n)!} \)
\(\frac{n+1}
{n-1}\)
\(\frac{(n-2)!}
{(n-1)!}\)
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?
\(\frac{11!}
{5!\, 4!\, 2!}=6\:930\)
\(5\cdot 4\cdot 2=40\)
\(5!\, 4!\, 2!=5\:760\)
\(\left (5!\, 4!\right )^{2}=8\:294\:400\)
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.
\(- 1\)
\(1\)
\(2\)
\(0\)