9000140510
Level:
B
Find the solution set of the equation assuming
\(x\in \mathbb{N}\).
\[
(x + 1)! + x! = 6x + 12
\]
\(\{3\}\)
\(\{2\}\)
\(\{1\}\)
\(\{4\}\)
Combinatorics
9000139309
Level:
A
There are \(20\) tablets in an
e-shop. From this amount \(18\)
tablets are new and \(2\)
tablets have been returned by customers. The e-shop manager gets an order
containing three tablets and he wants to get rid of the returned tablets first. How
many possibilities exist to complete the order?
\(18\)
\(\frac{18!}
{3!\; 15!}=816\)
\(18\cdot 16\cdot 3=864\)
\(20\cdot 19\cdot 18=6\:840\)
9000139303
Level:
A
A DJ's playlist contains \(18\)
songs. In this list there are \(7\)
rap songs, \(5\)
oldies and \(6\)
rock songs. The opening part should consist of one rap song, two oldies and one rock
song. The order of the songs does not matter. Find the number of possible ways how
to put the opening together.
\(420\)
\(120\)
\(320\)
\(520\)
9000139310
Level:
A
There are \(20\) tablets in an
e-shop. From this amount \(18\)
tablets are new and \(2\)
tablets have been returned by customers. The e-shop manager gets an order
containing three tablets and he wants to use only the new tablets for this order. How
many possibilities exist to complete the order?
\(\frac{18!}
{3!\; 15!}\)
\(18\)
\(18\cdot 16\cdot 3\)
\(20\cdot 19\cdot 18\)
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.
\(- 1\)
\(1\)
\(2\)
\(0\)