Permutations
1. Define and characterize:
b) Permutations from n elements with repetition
Solution:
Permutations
A pemutation is a sequence containing each element from a finite set of n elements once, and only once. Permutations of the same set differ just in the order of elements.
P(n) = n!
Permutations with repetition 
n1 – # of the same elements of the first cathegory
n2 - # of the same elements of the second cathegory
n3 - # of the same elements of the third cathegory
nj - # of the same elements of the j-th cathegory
Stands: n1+n2+n3+.........+nj = n
2. There are 3 elements a, b, c. Use them to make:
b) permutations with repetition, where „ a“ occures 2 times,“ b“ one time and „c“ one time
3.How many positive integers of 5 digits may be made from the ciphers 1,2,3,4,5, if:
b) How many of them will begin with 5?
c) How many of them will be even?
a) P(5) = 5! = 5.4.3.2.1 = 120
b) P(4) = 4! = 4.3.2.1 =24
c)
ending with 2: P(4) = 4! =24
ending with 4: P(4) = 4! = 24
sum : S = 2.4! = 2.24 = 48
4.Eight students should be accomodated in two 3-bed and one 2-bed rooms. In how many ways can they be accomodated?
Please log in to view the solution.5.In how many ways can you throw a sum of 11 by 3 playing cubes?
Please log in to view the solution.6.There are 4 czech and 3 slovak books on the bookshelf. Czech books should be placed on the left side of the bookshelf and slovak books on the right side of the bookshelf. How many ways are there to arrange the books?
Please log in to view the solution.7. If the number of elements would raise by 2, the number of permutations without repetition would raise 42 times. How many elements are there?
8.Count all possible arrangements of 15 people on the foto.
b) How long would the photographing last if you need 10 seconds for one foto?
9.A password of 6 digits is made of digits 926002.
How many possible passwords are there?
How long would it take to try all the possible passwords if trying one password takes 5 seconds?
10. A student has to make 4 university entry exams. For passing each exam he gets either 2,3 or 4 points. He needs to reach at least 13 points to get to the university. In how many ways can he do the exams to be succesfull?
Please log in to view the solution.