2000004503
Level:
A
How many lines are determined by \(6\) different points in the plane, if any three of the points do not lie in one straight line?
\( {{6}\choose{2}} =15\)
\( \frac{6!}{2!} =360\)
\( {{6}\choose{3}} =20\)
\( 6\)
Combinatorics
2000004504
Level:
A
They have \(6\) types of pants and \(5\) types of T-shirts in the sports store. In how many ways can we pick \(1\) pants and \(1\) T-shirt?
\( 6 \cdot 5 = 30\)
\( {{11}\choose{2}} = 55\)
\( {{6}\choose{2}} \cdot {{5}\choose{2}} =150\)
\( 6! \cdot 5! = 86\,400\)
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.
2010007001
Level:
A
There are \(8\) different books in the bookcase. In how many ways can they be arranged side by side?
\(8!=40\:320\)
\(8\)
\( 8^2=64\)
\(8^8=16\:777\:216\)
2010007002
Level:
A
The seven-digit bank vault code is made from the same numbers as \(9926002\). How many options are there to create the appropriate code?
\(\frac{7!}
{(2!)^3}=630\)
\(7!=5\:040\)
\(\frac{7!}
{2 \,\cdot\, 2!}=1\:260\)
\(\frac{7!}
{3!\, 2!}=420\)
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?
\(\frac{6!}
{4!\, 2!}\cdot \frac{8!}
{4!\, 4!}=1\:050\)
\(\frac{6!}
{4!}\cdot \frac{8!}
{4!}=50\:400\)
\(2\cdot 4=8\)
\(6\cdot 8=48\)
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?
\( 26^3 \cdot 10^4\)
\( 10^3 \cdot 26^4\)
\(36^7\)
\(26\cdot 25\cdot 24\cdot 10^4\)
2010007006
Level:
A
Find the number of possible ways how to organize a group of \(4\) boys and \(6\) girls in one ordered row, if all the boys should stand on the first four positions and the girls on remining positions.
\( 4! \cdot 6!\)
\( 10!\)
\( 4 \cdot 6\)
\(\frac{10!}
{4!\, 6!}\)
2010007104
Level:
A
There are \(5\) different roads between cities A and B. Find the number of possible ways from the city A to the city B and back, if it is required to use one road from A to B and another different one from B to A.
\( 5 \cdot 4 = 20\)
\( 5 + 4 = 9\)
\( 5 \cdot 5 = 25\)
\( 2 \cdot 5 = 10\)
2010007105
Level:
A
There are \(20\) girls and \(10\) boys in the class. How many ways are there to designate a president and vice-president of the class if it is required that at least one position will be held by a girl.
\(2\cdot 20\cdot 10 + 20 \cdot 19 =780\)
\(2\cdot 20\cdot 10=400\)
\(20\cdot 19 =380\)
\(20\cdot 10 =200\)