Combinatorics

9000148909

Level: 
A
There are \(24\) girls and \(8\) boys in the class. How many ways are there to designate a president and vice-president of the class if it is required that one of the position will be held by a boy and the other one by a girl?
\(24\cdot 8\cdot 2=384\)
\(24\cdot 8=192\)
\(\frac{32!} {2!\; 30!}=496\)
\(\frac{32!} {24!\; 8!}=10\:518\:300\)

9000148904

Level: 
A
Pamela needs new ski for a ski course. There are skis from six different vendors in a shop. The shop has four different ski pairs from each vendor, but two vendors have all products behind Pam's financial limit. How many pairs are at disposal for Pam?
\(4\cdot 4=16\)
\(4!=24\)
\(4\cdot 2=8\)
\(4 + 2=6\)

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.)
\(4\)
\(5\)
\(2\)