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Combinatorics

Probability : Permutations and Probability Distributions

1. You dream of someday winning the lottery (don't we all). You found out about a new lottery where the winning numbers are five different numbers between 1 and 34 inclusive. To win the lottery, you must select the correct 5 numbers in the same order in which they were drawn. According to your calculations, the probability of wi

Interpretation and Comparison of Combination Permutation

1. This week we were introduced to new terminology and symbols. Please interpret the symbol P(B|A) and explain what is meant by the expression. Why is P(B|A) not the same as P(B)? 2. Consider the formulas: nPr =n!/(n-r)! and nCr = n!/(n-r)!r! a. Given the same values for n and r in each formula, which is the smaller val

Sets and counting

Six people are going to travel to Mexico City by car. There are six seats available in the car. In how many different ways can the six people be seated in the car if only three of them can drive?

Combinations : 'Pick k out of n' Problem

Baskin Robbins serves "31 Flavors" of ice cream. How many different two-scoop ice cream cones are possible if two different flavors are used on each cone and the order of the scoops doesn't matter?* (* Means chocolate on top and vanilla on the bottom is the same as chocolate on the bottom and vanilla on top.)

Sets : One-to-one Correspondence

A.) Show that the following set is infinite by setting up a one-to-one correspondence between the given set and a proper subset of itself: {8,10,12,14,...} b.) Show that the following set has cardinal N sub o by setting up a one-to-one correspondence between the set of counting numbers and the given set: {5,9,13,17,...}

A Set D is a Subset of Set C

A set D is a subset of set C provided that? a. every element of C is an element of D b. every element of D is an element of C c. at least one element of C is not an element of D d. at least one element of D is not an element C e. none of the above.

Combinations-Number of choices

1. Joey is having a party. He has 10 friends, but his mom told him he could only invite 6 of them. How many choices are there if a. there are no restrictions b. there are 2 brothers who will only attend if they can attend together c. there are 2 girls who each will not attend if the other one does

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,

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.

Important Information about Counting

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

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

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.

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)

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

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

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 ,

Combinations, Permutations and Truth Tables

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

Combinations : Seating Arrangements at a Dinner Table

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?

Permutations and combinations

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

Number Patterns : Door Problem

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,

Combinations and Permutations: Sequential Counting Principle

An airline has 15 flights from city A to city B and eight flights from city B to city C. In how many ways can you fly from city A to city B. that is one type of problem another type is Karen has Five rabbits: 2 black, 2 white, 1 black and white. In how many ways can Karen select two of her rabbits and include at least one bla