2000004503
Level:
A
How many lines are determined by \(6\) different points in the plane, if any three of the points do not lie in one straight line?
\( {{6}\choose{2}} =15\)
\( \frac{6!}{2!} =360\)
\( {{6}\choose{3}} =20\)
\( 6\)
Combinatorics
2000004502
Level:
A
Specify the number of all two-element subsets of the set \( \{A, B, C, D, E \}\).
\( {{5}\choose{2}} =10\)
\( \frac{5!}{2!} =30\)
\( \frac{5!}{3!} =20\)
\( 5\)
2000004501
Level:
A
How many ways are there to arrange the letters in the word DOG?
\( 3! = 6\)
\(3\)
\(5\)
\(4\)
2000002806
Level:
B
Simplify for \( n \in \mathbb{N}\):
\[\ \log_{10}{[(n+3)!]}-\log_{10}{[(n+2)!]}\]
\( \log_{10}{(n+3)}\)
\( \log_{10}{(n+2)}\)
\( \log_{10}{\frac{n+3}{n+2}}\)
\( n+3\)
2000002805
Level:
B
Specify the domain of the expression:
\[\frac{(n+4)!}{(n+5)!}\]
\( n \in \mathbb{Z},~n\geq -4\)
\( n \in \mathbb{Z},~n\geq -5\)
\( n \in \mathbb{N}\)
\( n \in \mathbb{Z},~n\leq -5\)
2000002804
Level:
B
Specify the domain of the expression:
\[\frac{2\cdot (n-4)!}{(n-1)!}\]
\( n \in \mathbb{N},~n \geq 4 \)
\( n \in \mathbb{N},~n \geq 5 \)
\( n \in \mathbb{N} \)
\( n \in \mathbb{N},~n \geq 2 \)
2000002803
Level:
B
Simplify for \( n \in \mathbb{N}\):
\[\frac{2\cdot n!}{(n-1)!}\]
\( 2n\)
\( 2\cdot n!\)
\(\frac{2}{n-1}\)
\(\frac{2}{(n-1)!}\)
2000002802
Level:
B
Simplify for \( n \in \mathbb{N}\):
\[\frac{n!\cdot n!}{(n-1)!\cdot (n+1)!}\]
\( \frac{n}{n+1}\)
\( n^2 \)
\( \frac{n}{n-1}\)
\( \frac{1}{n-1}\)
2000002801
Level:
B
Simplify for \( n \in \mathbb{N}\) and \(n\geq 3\):
\[\frac{(n-3)!}{(n-2)!}\]
\(\frac{1}{n-2}\)
\( n-3\)
\(n-2\)
\(\frac{1}{n-3}\)
2000002706
Level:
B
Specify the domain of the expression: \(\left({n\above 0.0pt
5} \right)+\left({n+1\above 0.0pt
1} \right)\)
\( n \in \mathbb{N},~n\geq 5\)
\( n \in \mathbb{N},~n\leq 5\)
\( n \in \mathbb{N},~n\geq 6\)
\( n \in \mathbb{N},~n\geq 4\)