Combinatorics

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?
\(20\cdot 9^{3}\, \mathrm{s}=14\:580\,\mathrm{s}\)
\(20\cdot \frac{9!} {6!}\, \mathrm{s}=10\:080\,\mathrm{s}\)
\(20\cdot \frac{9!} {3!\, 6!}\, \mathrm{s}=1\:680\,\mathrm{s}\)
\(20\cdot 9\cdot 3\, \mathrm{s}=540\,\mathrm{s}\)

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?
\(24\cdot 8\cdot 2=384\)
\(24\cdot 8=192\)
\(\frac{32!} {2!\, 30!}=496\)
\(\frac{32!} {24!\, 8!}=10\:518\:300\)

9000141508

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