Combinatorics

9000141508

Level: 
B
Assuming \(x\in \mathbb{N}\), find the solution set of the following equation. \[ \left({x\above 0.0pt x}\right) +\left ({x + 1\above 0.0pt x} \right) +\left ({x + 2\above 0.0pt x} \right) +\left ({x + 3\above 0.0pt x} \right) = \frac{x^{3} + 59} {6} \]
\(\{1\}\)
\(\{4\}\)
\(\{10\}\)

9000141502

Level: 
B
Let \(A\) be set with \(n\) mutually different elements. The number of \(5\)-permutations with repetition is \(1024\). Find \(n\). (The term „\(k\)-permutation with repetition” stands for an ordered arrangement of \(k\) objects from a set of \(n\) objects, when each object can be chosen more than once.)
\(4\)
\(5\)
\(2\)

9000139710

Level: 
C
The wallet contains nine coins: three \(1\)-Euro coins, three \(2\)-Euro coins and three \(5\)-Euro coins. How many different amounts can be paid if we have to pay the amount exactly and use just three coins for this payment?
\(\frac{5!} {3!\, 2!}=10\)
\(\frac{5!} {3!}=20\)
\(3^{3}=27\)
\(3!=6\)

9000139703

Level: 
A
The box contains \(5\) red crayons, \(4\) yellow crayons and \(2\) green crayons. The crayons are removed from the box and arranged in a line. How many different color patterns can be obtained by this procedure?
\(\frac{11!} {5!\, 4!\, 2!}=6\:930\)
\(5\cdot 4\cdot 2=40\)
\(5!\, 4!\, 2!=5\:760\)
\(\left (5!\, 4!\right )^{2}=8\:294\:400\)