Combinatorics

9000148901

Level: 
A
The current Czech vehicle registration plate number has the form NLN-NNNN, where N stands for a digit from \(0\) to \(9\) and L stands for a letter from an alphabet containing \(26\) letters. How many different registration plates are possible?
\(26\cdot 10^{6}\)
\(10^{6}\)
\(15\cdot 10^{6} + 6\cdot 10^{5}= 156\cdot 10^{5}\)
\(16\cdot 10^{6}\)

9000148903

Level: 
A
A combination lock will open if a right choice of three numbers (from \(1\) to \(9\)) is selected. Suppose that we use a brute force attack to open the lock (we try all possibilities). To try one code takes \(20\) seconds. What is the maximal time (in seconds) required to open the lock by brute force?
\(20\cdot 9^{3}\, \mathrm{s}=14\:580\,\mathrm{s}\)
\(20\cdot \frac{9!} {6!}\, \mathrm{s}=10\:080\,\mathrm{s}\)
\(20\cdot \frac{9!} {3!\; 6!}\, \mathrm{s}=1\:680\,\mathrm{s}\)
\(20\cdot 9\cdot 3\, \mathrm{s}=540\,\mathrm{s}\)

9000148904

Level: 
A
Pamela needs new ski for a ski course. There are skis from six different vendors in a shop. The shop has four different ski pairs from each vendor, but two vendors have all products behind Pam's financial limit. How many pairs are at disposal for Pam?
\(4\cdot 4=16\)
\(4!=24\)
\(4\cdot 2=8\)
\(4 + 2=6\)

9000148907

Level: 
A
A bowl contains \(12\) different gummy-bears and \(20\) different sweet-drops. Anne can choose either one sweet-drop or one gummy-bear. From the rest, Jane can choose one sweet-drop and two gummy-bears. Anne wants to provide a maximum of the possibilities for Jane's choice. What should Anne choose?
sweet-drop
gummy-bear
Both possibilities give the same result.

9000148908

Level: 
A
There are seven different yellow apples, eight different green apples and ten different red apples. How many ways are there to choose three apples, if we wish to have three apples of different colors?
\(10\cdot 8\cdot 7=560\)
\(\frac{10\cdot 8\cdot 7} {2}=280 \)
\((10 + 8 + 7)\cdot 2=50\)
\(10 + 8 + 7=25\)

9000148909

Level: 
A
There are \(24\) girls and \(8\) boys in the class. How many ways are there to designate a president and vice-president of the class if it is required that one of the position will be held by a boy and the other one by a girl?
\(24\cdot 8\cdot 2=384\)
\(24\cdot 8=192\)
\(\frac{32!} {2!\; 30!}=496\)
\(\frac{32!} {24!\; 8!}=10\:518\:300\)