9000153907
Level:
C
Find the number of ways how to distribute
\(5\) different
balls among \(8\)
persons.
\(8^{5} = 32\:768\)
\(\left({12\above 0.0pt
5} \right) = 792\)
\(\frac{8!}
{3!} = 6\:720\)
\(\frac{8!}
{5!3!} = 56\)
Combinatorics
9000153906
Level:
C
Find the number of ways how to distribute
\(5\) identical
balls among \(8\)
persons so that no persons gets more than one ball.
\(\frac{8!}
{5!3!} = 56\)
\(\frac{8!}
{3!} = 6\:720\)
\(\left({12\above 0.0pt
8} \right) = 495\)
\(\left({12\above 0.0pt
5} \right) = 792\)
9000153804
Level:
A
Find the number of \(3\)-combinations
from the set \(\{2,3,4,5\}\).
\(4\)
\(6\)
\(24\)
\(20\)
9000153805
Level:
A
Find the number of \(3\)-combinations
with repetition from the set \(\{2,3,4,5\}\).
\(20\)
\(4\)
\(24\)
\(12\)
9000153903
Level:
C
Find the number of ways how to distribute
\(8\) mutually different
balls among \(5\)
persons.
\(5^{8} = 390625\)
\(\frac{8!}
{3!} = 6720\)
\(8^{5} = 32768\)
\(\left({12\above 0.0pt
8} \right) = 495\)
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?
\(20\cdot 9^{3}\, \mathrm{s}=14\:580\,\mathrm{s}\)
\(20\cdot \frac{9!}
{6!}\, \mathrm{s}=10\:080\,\mathrm{s}\)
\(20\cdot \frac{9!}
{3!\; 6!}\, \mathrm{s}=1\:680\,\mathrm{s}\)
\(20\cdot 9\cdot 3\, \mathrm{s}=540\,\mathrm{s}\)
9000148905
Level:
A
The king has eight daughters. A dragon asks two daughters, otherwise he will
destroy the whole country. In how many ways can we choose a pair of king's
daughters for the dragon?
\(\frac{8!}
{2!\; 6!}=28\)
\(8\cdot 7=56\)
\(8^{2}=64\)
\(8 + 7=15\)
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?
\(10\cdot 8\cdot 7=560\)
\(\frac{10\cdot 8\cdot 7}
{2}=280 \)
\((10 + 8 + 7)\cdot 2=50\)
\(10 + 8 + 7=25\)
9000148902
Level:
A
There are four paths from a city to the top of nearby mountain. Find the number of
possible treks from the city to the mountain and back, if it is required to use one
path up and another one down.
\(4\cdot 3=12\)
\(4\cdot 4=16\)
\(4 + 3=7\)
\(2\cdot 4=8\)