9000140509
Level:
B
Find the solution set of the equation assuming
\(x\in \mathbb{N}\).
\[
(x + 1)! = 6(x - 1)!
\]
\(\{2\}\)
\(\{ - 3;\ 2\}\)
\(\left \{\frac{5}
{7}\right \}\)
\(\left \{\frac{7}
{5}\right \}\)
Combinatorics
9000139306
Level:
A
How many different ice cream bowls can be prepared from
\(8\) kinds
of an ice cream if the bowl contains just three different kinds of ice cream?
\(\frac{8!}
{3!\; 5!}=56\)
\(\frac{8!}
{2!\; 5!}=168\)
\(\frac{8!}
{5!}=336\)
\(8!=40\:320\)
9000140510
Level:
B
Find the solution set of the equation assuming
\(x\in \mathbb{N}\).
\[
(x + 1)! + x! = 6x + 12
\]
\(\{3\}\)
\(\{2\}\)
\(\{1\}\)
\(\{4\}\)
9000139309
Level:
A
There are \(20\) tablets in an
e-shop. From this amount \(18\)
tablets are new and \(2\)
tablets have been returned by customers. The e-shop manager gets an order
containing three tablets and he wants to get rid of the returned tablets first. How
many possibilities exist to complete the order?
\(18\)
\(\frac{18!}
{3!\; 15!}=816\)
\(18\cdot 16\cdot 3=864\)
\(20\cdot 19\cdot 18=6\:840\)
9000139303
Level:
A
A DJ's playlist contains \(18\)
songs. In this list there are \(7\)
rap songs, \(5\)
oldies and \(6\)
rock songs. The opening part should consist of one rap song, two oldies and one rock
song. The order of the songs does not matter. Find the number of possible ways how
to put the opening together.
\(420\)
\(120\)
\(320\)
\(520\)
9000139310
Level:
A
There are \(20\) tablets in an
e-shop. From this amount \(18\)
tablets are new and \(2\)
tablets have been returned by customers. The e-shop manager gets an order
containing three tablets and he wants to use only the new tablets for this order. How
many possibilities exist to complete the order?
\(\frac{18!}
{3!\; 15!}\)
\(18\)
\(18\cdot 16\cdot 3\)
\(20\cdot 19\cdot 18\)
9000139304
Level:
A
Find the number of possibilities how to choose a pair of timekeepers for a sport event if
there are \(50\)
candidates available for this job.
\(\frac{50!}
{2!\; 48!} = 1\:225\)
\(\frac{50}
{2} = 25\)
\(50^{2} = 2\:500\)
\(\frac{50!}
{48!} = 2\:450\)
9000139701
Level:
A
There are \(15\)
athletes in an athletic meeting. Determine in how many ways it is possible to
obtain the results on the first six places of the scoreboard if the place on
scoreboard cannot be shared (one athlete per one place on scoreboard).
\(\frac{15!}
{9!} =3\:603\:600\)
\(6^{15}=470\:184\:984\:576\)
\(15!\, 6!=941\:525\:544\:960\:000\)
\(\frac{15!}
{9!\, 6!}=5\:005\)
9000139305
Level:
A
There are five rooms with three beds and one room with five beds in a hotel. A group of
\(20\)
people booked rooms at this hotel. Determine the number of possible choices for the
people to the five-bed room.
\(\frac{20!}
{5!\; 15!}=15\:504\)
\(20\cdot 3\cdot 5=300\)
\(\frac{20!}
{3!\; 5!}=3\:379\:030\:566\:912\:000\)
\(20^{5}=3\:200\:000\)
9000139702
Level:
A
A race was attended by \(12\)
competitors. Find the number of possible orderings of the competitors on the first
three places of the scoreboard.
\(\frac{12!}
{9!} =1\:320\)
\(3^{12}=531\:441\)
\(\frac{12!}
{9!\, 3!}=220\)
\(12!\, 3!=2\:874\:009\:600\)