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# Combinatorics

### Combinations in Committees and Groups

5. In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each? 6. From a group of n people, suppose that we want to choose a committee of k, k <= n, one of whom is to be designated as a chairperson. (a) By focusing first on the choice of the committee and then

### Permutations and Combinations of Committees

We wish to form a committee of 7 people chosen from 5 democrats, 4 republicans, and 6 independents. The committee will contain 2 democrats, 2 republicans, and 3 independents. In how many ways can we choose the committee?

### Permutation Groups : Cycles

Here's my problem: Let (i1, i2, . . . , ik) be a k-cycle (k less or equal to n) element of Sn and let sigma be an element of Sn. (i) Find a precise expression for sigma * (i1, i2, . . . , ik)* sigma-inverse. Hint: experiment a little, perhaps, then take a guess and prove it. (ii) Describe precisely the set {sigma * (1,

### Fields, Elements and Cyclic Groups

Find H K in {see attachment}, if H = <|3|> and K = <|5|>. This is all the problem says. I know the answer, but I do not know the reasoning.

### Permutation Subgroups

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

### Permutation Groups - Rigid Motion of a Cube

A rigid motion of a cube can be thought of either as a permutation of its 8 vertices or as a permutation of its 6 sides. Find a rigid motion of a cube that has order 3, and express the permutation that represents it in both ways, as a permutation on 8 elements and as a permutation on 6 elements.

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

### Discrete Math : Permutations

At the Clone Zone School a class of 10 identical boys and 8 identical girls go to a cafeteria. There are three registers : A, B and C. In how many ways can the students line up at the three registers?

### Introductory probability, basic combination/permutation

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

### Partitions on a Set

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

### Counting

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.

### Linear Congruences Set of Mutually Incongruent Solutions

Find a complete set of mutually incongruent solutions of each of the following. (a) 7x is congruent to 5 (mod 11) (b) 8x is congruent to 10 (mod 30) (c) 9x is congruent to 12 (mod 15)

### Probability

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

### Characterizing the metric space {N}

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

### Permutations and Restrictions

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)

### Are combinations are just an application of the counting principle?

Are combinations are just an application of the counting principle?

### Combinations: 8 People 3 Seats

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

### Combinations : How Many Outfits?

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

### Lebesgue Set: Change of Variable Theorem and Integrability

|mA| = |detm| |A| For every Lebesgue set A belogning to R^n and every invertible n x n matrix m. Use this to prove the change of variable theorem. (*) The integral (mb) of f(y)dy = the integral (b) of f(mx)|detm| dx for everyy Lebesgue measurable function f = R^n --> R which is LEbesgue integrable on a Lebesgue set B the uni

### Combinations : Finding possible ways a social security number can be arranged.

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?

### Finance : Combinations, Interest, Annuities and Loans

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

### Combination & Permutation

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

### Combinations and Permutations : Six Problems

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

### Permutations Define Functions

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.

### Permutations : Disjoint Transpositions

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.

### Permutations : Parity Functions

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

### Combination: How Many Four-Digit Numbers Can Be Formed

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.

### Inversions of Permutation Found

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

### Order of a Permutation

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 ,

### Element of Finite Order

Prove that every element of Q/Z has finite order, where Q is the set of rational numbers with group operation + and Z is the set of integers.