Combinatorics

9000139302

Level: 
A
The phone number contains nine digits. A witness does not remember the full number, but he remembers that the phone number starts by \(728\), ends by \(01\) and there is no repeating digit in the number. How many phone numbers meet these conditions?
\(120\)
\(320\)
\(520\)
\(720\)

9000139308

Level: 
A
The shooting club has \(25\) members. Among the members it is necessary to vote a board: a president, a cashier and a webmaster. One person cannot have more than one of these positions and there is only one member skilled enough to be a webmaster. How many possibilities exist to set up the board?
\(24\cdot 23=552\)
\(25\cdot 24=600\)
\(24\cdot 23\cdot 22=12\:144\)
\(25\cdot 24\cdot 23=13\:800\)

9000136901

Level: 
B
The sum \(\left({15\above 0.0pt 8} \right) +\left ({15\above 0.0pt 9} \right)\) equals to:
\(\left({16\above 0.0pt 9} \right)\)
\(\left({15\above 0.0pt 10}\right)\)
\(\left({15\above 0.0pt 7} \right)\)
\(\left({16\above 0.0pt 8} \right)\)
\(\left({30\above 0.0pt 17}\right)\)

9000136905

Level: 
B
For \(n\in \mathbb{N}\), \(n\geq 2\), the difference \(\left({n\above 0.0pt 2} \right) -\left ({ n\above 0.0pt n-2}\right)\) equals to:
\(0\)
\(\left (n + 2\right )\left (n + 1\right )\)
\(\left({n+2\above 0.0pt n} \right)\)
\(n^{2} - 1\)
\(\left({n\above 0.0pt n}\right)\)

9000136903

Level: 
B
Simplify \(\left({4\above 0.0pt 0}\right) +\left ({4\above 0.0pt 1}\right) +\left ({4\above 0.0pt 2}\right) +\left ({4\above 0.0pt 3}\right) +\left ({4\above 0.0pt 4}\right)\).
\(4^{2}\)
\(14\)
\(\left({5\above 0.0pt 4}\right)\)
\(32\)
\(\left({8\above 0.0pt 4}\right)\)