A

1003024611

Level: 
A
On a lock of a safe deposit box a ten-digit code can be set. The code can consist only of four \( 1 \)s, three \( 2 \)s, two \( 3 \)s, and one \( 4 \). How many ways are there to set the code?
\( \frac{10!}{4!\cdot3!\cdot2!} = 12\:600 \)
\( \frac{10!}{4!+3!+2!}=113\:400 \)
\( 10!-4!\cdot3!\cdot5!=3\:628\:512 \)
\( 10! = 3\:628\:800 \)

1003024610

Level: 
A
In a high-speed train set, there should be the following cars included: \( 3 \) first class cars, \( 5 \) second class cars, \( 2 \) sleeping cars, \( 1 \) dining car, and \( 2 \) luggage cars. How many ways are there to arrange the cars in this high-speed train set?
\( \frac{13!}{(2!)^2\cdot3!\cdot5!}=2\:162\:160 \)
\( \frac{13!}{(2!)^2+3!+5!}=47\:900\:160 \)
\( 13!-(2!)^2\cdot3!\cdot5!=6\:227\:017\:920 \)
\( 13!-\left|(2!)^2+3!+5!\right|=6\:227\:020\:670 \)

1003024607

Level: 
A
On the shelf, there should be three blue cups, three red cups, two yellow cups and two green cups arranged in a row from the left to the right. The cups of the same color are not mutually distinguishable. How many arrangements of these cups are possible?
\( \frac{10!}{(2!)^2\cdot(3!)^2}=25\:200 \)
\( \frac{10!}{4\cdot6!}=1\:260 \)
\( \frac{10!}{2\cdot2!\cdot3!}=151\:200 \)
\( \frac{10!}{4\cdot2!\cdot3!}=75\:600 \)

1003024606

Level: 
A
Each payment card has its numeric four-digit PIN code. How many different PIN codes can be selected, if only a code with different numbers may be used?
\( \frac{10!}{6!} = 5\:040 \)
\( \frac{10!}{4!} = 151\:200 \)
\( \frac{10!}{6!\cdot4!} = 210 \)
\( 10^4 = 10\:000 \)

1003024601

Level: 
A
Assume the password for the safe deposit box consists of four different letters from the set \( \{A;B;C;D;E;F;G;H\} \) and four different numbers from the set \( \{1;2;3;4;5;6;7\} \). How many different passwords are there?
\( \binom84 \cdot \binom74 \cdot 8! = 98\,784\,000 \)
\( \frac{8!}{4!}\cdot\frac{7!}{3!}\cdot8!=56\,899\,584\,000 \)
\( \left(\frac{8!}{4!}+\frac{7!}{3!}\right)\cdot8! = 101\,606\,400 \)
\( \left(\binom84+\binom74\right)\cdot8!=4\,233\,600 \)

1003019103

Level: 
A
There are \( 30 \) students in the class, one of them is Adam. The teacher picks randomly three students to be tested. What is the probability that Adam is among them?
\( \frac{\binom{29}2}{\binom{30}3}=0{.}1 \)
\( \frac{\binom{29}2}{\binom{30}2}\doteq 0{.}9333 \)
\( \frac{\binom{29}3}{\binom{30}3}=0{.}9 \)
\( \frac{\binom31\binom{27}2}{\binom{30}{3}}\doteq 0{.}2594 \)