Combinatorics

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

9000141501

Level: 
B
Let \(A\) be set with \(n\) mutually different elements. If \(n\) is increased by \(2\), then number of \(3\)-permutations is increased by \(384\). Find \(n\). (The term „\(k\)-permutation” stands for an ordered arrangement of \(k\) objects from a set of \(n\) objects.)
\(8\)
\(64\)
\(32\)

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?
\(\frac{11!} {5!\, 4!\, 2!}=6\:930\)
\(5\cdot 4\cdot 2=40\)
\(5!\, 4!\, 2!=5\:760\)
\(\left (5!\, 4!\right )^{2}=8\:294\:400\)

9000139704

Level: 
C
There are \(5\) different kinds of cakes in a shop. Find the number of possibilities how to buy \(8\) cakes in this shop. (There is more than \(8\) cakes of each kind available.)
\(\frac{12!} {8!\, 4!}=495\)
\(5!\, 8!=4\:838\:400\)
\(5^{8}=390\:625\)
\(\frac{8!} {5!\, 3!}=56\)

9000139705

Level: 
A
From the group of \(10\) boys and \(5\) girls we have to select a small group of \(3\) boys and \(2\) girls. How many possibilities exist for this choice?
\(\frac{10!} {7!\, 3!}\cdot \frac{5!} {3!\, 2!}=1\:200\)
\(5^{10}=9\:765\:625\)
\(10\cdot 5!\, 3!=7\:200\)
\(5\cdot \frac{10!} {3!} =3\:024\:000\)