1003028802
Level:
A
Find the sum of all the real roots of the next equation. Multiple roots, if any, count only once.
\[ \left(x^2+9\right)\left(x^2-1\right)\left(x+1\right)=0 \]
\( 0 \)
\( 1 \)
\( -1 \)
\( 3 \)
A
1003028801
Level:
A
For the next equation, find the sum of all the roots from the set of natural numbers. Multiple roots, if any, count only once.
\[ \left(x^2+4\right)\left(x^2-9\right)=0 \]
\( 3 \)
\( -5 \)
\( 5 \)
\( -1 \)
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?
\( \frac{13!}{(2!)^2\cdot3!\cdot5!}=2\:162\:160 \)
\( \frac{13!}{(2!)^2+3!+5!}=47\:900\:160 \)
\( 13!-(2!)^2\cdot3!\cdot5!=6\:227\:017\:920 \)
\( 13!-\left|(2!)^2+3!+5!\right|=6\:227\:020\:670 \)
1003024609
Level:
A
How many \( 8 \)-digit numbers are there in which the digit sum is \( 3 \)?
\( 36 \)
\( 113 \)
\( 112 \)
\( 35 \)
1003024608
Level:
A
How many ways are there to arrange the letters in the Czech word KOMBINATORIKA?
\( \frac{13!}{(2!)^4}=389\:188\:800 \)
\( \frac{13!}{4\cdot2!}=778\:377\:600 \)
\( \frac{13!}{2!}=3\:113\:510\:400 \)
\( 13!-4\cdot2!=6\:227\:020\:792 \)
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?
\( \frac{10!}{6!} = 5\:040 \)
\( \frac{10!}{4!} = 151\:200 \)
\( \frac{10!}{6!\cdot4!} = 210 \)
\( 10^4 = 10\:000 \)
1003024602
Level:
A
Suppose six pupils are sitting on six chairs in a row. Among them there are twins. In how many ways can the pupils be seated in the row so that the twins do not sit next to each other?
\( 6! -2\cdot5!=480 \)
\( \frac{6!}2=360 \)
\( 2\cdot5!=240 \)
\( 6! -5!=600 \)
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?
\( \binom84 \cdot \binom74 \cdot 8! = 98\,784\,000 \)
\( \frac{8!}{4!}\cdot\frac{7!}{3!}\cdot8!=56\,899\,584\,000 \)
\( \left(\frac{8!}{4!}+\frac{7!}{3!}\right)\cdot8! = 101\,606\,400 \)
\( \left(\binom84+\binom74\right)\cdot8!=4\,233\,600 \)