Combinatorics

2010007103

Level: 
B
Assuming \(x\in \mathbb{N}\), \(n\geq 2\), find the solution set of the following inequality. \[ \left({ x\above 0.0pt x - 2}\right)\cdot \left({x\above 0.0pt 2}\right) - 20\cdot \left({x\above 0.0pt 2}\right) + 96 < 0 \]
\(\{5\}\)
\(\{9;10;11\}\)
solution does not exist
\( (8;12)\)

2010007102

Level: 
B
Consider a set of \(n\) mutually different objects. If \(n\) is increased by \(5\), the number of \(2\)-permutations of these objects is increased by \(340\). Find \(n\). (The term „\(k\)-permutation” stands for an ordered arrangement of \(k\) objects from a set of \(n\) objects.)
\( 32\)
\( 34\)
\( 64\)
\( 18\)

2010007005

Level: 
A
A license plate of a car consists of \(7\) symbols so that letters are on the first three positions and digits on remaining four positions, while any used symbol can be repeated. Letters are chosen from \(26\) symbols of the alphabet and digits are chosen from the set \(\{0; 1;\dots; 9\}\). How many such license plates can be set up?
\( 26^3 \cdot 10^4\)
\( 10^3 \cdot 26^4\)
\(36^7\)
\(26\cdot 25\cdot 24\cdot 10^4\)

2010007004

Level: 
A
From a group of \(6\) boys and \(8\) girls we have to select a small group of \(2\) boys and \(4\) girls. How many possibilities exist for this choice?
\(\frac{6!} {4!\, 2!}\cdot \frac{8!} {4!\, 4!}=1\:050\)
\(\frac{6!} {4!}\cdot \frac{8!} {4!}=50\:400\)
\(2\cdot 4=8\)
\(6\cdot 8=48\)

2000004505

Level: 
A
Out of \(15\) boys and \(15\) girls in a class, \(5\) boys and \(5\) girls got A, \(5\) boys and \(5\) girls got B, and another \(5\) boys and \(5\) girls got C on a Math-test. (There were no Ds and Fs on this test.) Determine the smallest value of \(n\in\mathbb{N}\), so that if a team of \(n\) children is set-up, then certainly at least two children of the same gender and with the same grade are on the team.
\( 7\)
\( 6\)
\( 15 \)
Could not be determined.