### Permutation groups

Show that in any group of permutations, the set of all even permutations forms a subgroup.

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Show that in any group of permutations, the set of all even permutations forms a subgroup.

Eight people are attending a seminar in a room with eight chairs. In the middle of the seminar, there is a break and everyone leaves the room. a) In how many ways can the group sit down after the break so that no-one is in the same chair as before? b) In how many ways can the group sit down after the break so that exactly

There is a lottery in which 2000 individuals enter, and of these a set of 120 names will be randomly selected. Assume that both you and your friend are entered in the lottery. a. In how many ways can 120 names be randomly selected from the 2000 in the drawing? b. In how many ways can the drawing be done in such a way that

We denote the number of partitions of a set of n elements by P(n). Suppose the number of partitions of a set on n elements into k parts is denoted by P(n,k). Then obviously P(n) = P(n,1) + P(n,2) + ..... + P(n,n) Show that P(n,2) = 2^(n-1) - 1

A. An office manager has four employees and nine reports to be done. In how many ways can the reports be assigned to the employees so that each employee has at least one report to do. b. Find the number of ways to put eight different books in five boxes, if no box is allowed to be empty.

An automobile license number contains 1 or 2 letters followed by a 4 digit number. Compute the maximum number of different licenses.

For the metric space { N }, the set of all natural numbers, characterize whether or not it has the following properties: compact, totally bounded, has the Heine-Borel property, complete. For compact, we are to show that every sequence converges. For totally bounded, we are to show that it can be covered by finitely many sets

Can you check my answers and help me with B? Preparing a plate of cookies for 8 children, 3 types cookies {chocolate chip, peanut butter, oatmeal}, unlimited amount of cookies in supply but only cookie per child. One cookie per plate, one plate per child. A) How many different plates can be prepared? C(8,3) = 56 B)

This is a test on Sets, Functions and Permutations. There are several questions. Please see the attached file for the fully formatted problems.

Are combinations are just an application of the counting principle?

There are 8 people and 3 seats. How many different ways can they be seated.

Six pair of jeans, 3 shirts, and 2 pair of sandals. How many outfits can you have?

1. A social security number has 9 digits. If numbers can be repeated and all numbers can be used, there would be 10^9 possible social security numbers. Find each of the following. a. How many possible social security numbers are there if numbers can be repeated but 0 cannot be used for the first digit?

1) An admissions test given by a university contains 10 true-false questions. Eight or more of the questions must be answered correctly in order to be admitted. a) How many different ways can the answer sheet be filled out? b) How many different ways can the answer sheet be filled out so that 8 or more questions are answere

How many ways can you select a committee of 4 men from among 8 people?

1) Give clearly justified answers to the following. a) How many 7-digit telephone numbers can be formed if the first digit cannot be 0 or 9 and if the last digit is greater than or equal to 2 and less than or equal to 3? Repeated digits are allowed. b) How many different ways are there to arrange the 6 letters of the word CA

Define the sign of a permutation $ to be: sgn $ = 1 if $ is even. -1 if $ is odd. Prove that sgn($%) = sgn$sgn% for all $ and % in Sn.

If $ belongs to Sn (where Sn is the symmetric group of degree n), show that $^2 = % if and only if $ is a product of disjoint transpositions.

Show that $ and %$%^-1 have the same parity for all % and $ in Sn. (Sn is the symmetric group of degree n)

Q: How many four-digits numbers can be formed under the following conditions? (a) Leading digits cannot be zero. (b) Leading digits cannot be zero and no repetition of digits is allowed. (c) Leading digits cannot be zero and the number must be a multiple of 5.

Find the number of inversions in the following permutation: (3, 2, 1)

Please see the attached file for the fully formatted problems. Find the order of sigma^1000, where sigma is the permutation (123456789) (378945216) Find the order of , where is the permutation . Solution. Since and . Let ,

Show that if a,b is an element of Q then a+b is an element of Q and ab is an element of Q.

1. (a) How many license plates can a state produce if the plates can contain 6 characters (from 26 letters and 10 digits) if they can only use one digit? (b) How many ways can Mr. Paul choose 6 students from a class of 15 Boys and 12 Girls, if he must choose at least 5 boys? (c) How many orderings are there of the letter

Find the number of different selections of three letters which can be made from the letters of the word PARALLELOGRAM. How many of these contain the letter P?

In how many ways can 6 couples be seated at a circular table if each couple is not to be separated? How many ways can 5 Manchester United and 8 Chlesea players be seated at a circular dinner table if no two Manchester United players can sit together?

A set of 10 flags, 5 red, 3 blue and 2 yellow are to be arranged in a line along a balcony. If flags of the same colour are INDISTINGUISHABLE, find the number of arrangements in which, 1) The three blue flags are together 2) The yellow flags are not together 3) The red flags occupy alternate positions in the line 4) If the

A school had a very unusual tradition involving its 1000 students and its 1000 lockers. On opening day, after the head of the school had closed all the lockers, a student walked by and opened every single one. A second student then closed every second one (#2, 4, 6, 8 etc). A third student then changed every third locker (#3,

How many 5 card poker hands are there? Note a deck of 52 cards is assumed.

For each new employee, a company gives a five-digit identification card. Each digit can be 0, 1, 2, or 3. If repetitions are allowed, how many different cards are possible. A. 1024 B. 625 C. 25 D. 500