Combinatorics(matura)

The five graduating classes in a school perform 1 dance each at the graduation ball. The class A perform the opening dance, a “palotás”. The order of the other dances is decided by a draw. How many different orders are possible?

The lateral faces of a pyramid are painted in six different colours, and each face is painted in a single colour. How many different colourings are possible? (Two colourings are considered different if they cannot be obtained from each other by rotating the pyramid.)

Four different kind of fruit trees are planted in a row: an apple tree, a pear tree, a peach tree and a plum tree. It is also given that there can be no peach tree at either end of the row. How many ways are there to plant these trees?

Three digit numbers are formed using the digits 1, 2, 3, 4, 5 in such a way that non of them contains equal digits. How many numbers can be formed this way?

How many 4-digit positive integers exist in the decimal system that have four different odd digits?

How many (positive) three-digit odd numbers are there in the decimal system that consist of 3 different digits?

How many four-digit numbers of four different digits can be made, such that each digit is an element of the set {1; 2; 3; 4; 5; 6; 7} ?

How many seven-digit numbers divisible by four can be made out of digits of 1, 2, 3, 4, 5?

How many six-digit numbers divisible by three are there that only contain digits of 1, 2, 3, 4, 5, such that each of these digits occurs at least once?

In a test, students wrote three-letter codes made up of the letters A, B and C on their papers instead of their names. Every possible code from AAA to CCC was given to a student, and there were no students with the same code. How many students took the test?