Combinatorics

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?
\(\frac{10!} {7!\, 3!}\cdot \frac{5!} {3!\, 2!}=1\:200\)
\(5^{10}=9\:765\:625\)
\(10\cdot 5!\, 3!=7\:200\)
\(5\cdot \frac{10!} {3!} =3\:024\:000\)

9000139706

Level: 
A
The international alphabet contains \(26\) letters. The letters of this alphabet and the digits from \(0\) to \(9\) are used to form a code of the length \(4\) (a code contains \(4\) characters). The characters may repeat through the code and the code is not case sensitive (uppercase letters are equivalent to lowercase letters). How many codes can be obtained?
\(36^{4}=1\:679\:616\)
\(10\cdot 26^{4}=4\:569\:760\)
\(\frac{36!} {32!\, 4!}=58\:905\)
\(\frac{26!} {22!\, 4!}=14\:950\)

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?
\(120\)
\(320\)
\(520\)
\(720\)