Combinatorics Definition and 421 Threads

(I apologize for the following lack of clarity, I know next to nothing. Links would be cool if the answer would take to long, I simply don't know how to even google these questions ) Question 1, non decimal numbers: I have a set of parameters I'd like to encode. Each parameter is in the form...

Combinatorics is a branch of mathematics that study enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize them. Thus, the topics of Permutation and Combination come under Combinatorics. They refer to the related problem of counting the...

While bored in class one day I started to come up with a problem that I kept making more difficult as I solved each step. The overall setup goes like this. Picture an alarm clock or a scoreboard clock. There are 5 sections of the displayed time. 4 places are for where numbers go and the middle...

Hi Everyone, Homework Statement If we are asked the number of ways 2n people can be divided into 2 groups of n members, can I first calculate the number of groups of n members that can be formed from 2n people and then calculate number of ways 2 groups can be selected from the number of groups...

Homework Statement Consider f:[7]→[9] How many functions are there in which f^-1 is not a function? Homework Equations Don't seem to fine one. The Attempt at a Solution Try drawing the function diagram, given f:[7]→[9]. Then, count the total number of functions. Take too long to do so...

Let's say we have a deck with 40 cards. There are two of each for each of the 4 suits: 10, Jack, Queen, King, and Ace. Each hand consists of 10 cards. Given that each pair is technically the same (one 10 of hearts is not distinguishable from the other 10 of hearts), how would one calculate the...

How many straight poker hands are there (not straight flush) from a deck of 51 cards with the queen of spades missing? A is high or low. I know the answer is 9435, but I want to know why. I'm not sure how to approach this problem. Any help?

Homework Statement For any positive integer n determine: \sum\limits^n_{i=0} \frac{1}{i!(n-i)!}Homework Equations I don't really know where to start.. Up until this point we've just been doing permutations, combinations, and determining the coefficient of a certain term in the expansion of a...

Watch the image below. If we combine the two triangles we get different results. Triangles will be replaced with the number 3 (because triangles have three angles), the results obtained with the number as a geometric object angles. Connecting the two triangles is the mathematical operations of...

Suppose you have written down a seven digit phone number, but you realize you made a mistake and two of the digits are interchanged. For example, suppose the correct number is 345-6789, but you have on accident written down 348-6759. Sticking to the general case though, where you have seven...

This is a question taken from the AMC 2012 12 B exam held in February. I did not answer it during the exam, but now I try to complete all of it at home. I thought I found a solution, but it is not in the proposed choices, and so I am really lost... And the solution is not available for this...

Hello, I am having trouble solving this problem. Maybe I'm just overreacting to it. In my two semesters in discrete math/combinatorics, I've never seen a problem like this (with two summations) and been asked to prove it. Can some one help? \sum^{n}_{i=1} i^3 = \frac{n^2(n+1)^2}{4} =...

Combinatorics Cameron -- Lucas' Theorem Proof Hi everybody -- Im currently going through Peter Cameron's combinatorics book, and I'm having trouble understanding a step in the proof of Lucas' Theorem, given on page 28 for those of you with the book. The theorem states for p prime, m =...

I am a first-year physics major (currently in calc 2) and was wondering if there are any good introductory/low-level books on combinatorics through which I could educate myself over the summer. My college is offering a topics course on it next semester, but I was told by the professor that it...

Hi, My question is related to a math puzzle. The puzzle asks to swap any two digits from the numerator with any two digits in the denominator of the fraction 1630 / 4542 to get the ratio 1 / 3. Two possible answers are 1534 / 4602 and 1354/ 4062. What I am struggling with is the number...

Ques:A boat is to be manned by eight men, of whom 2 can only row on bow side and 1 can only row on stroke side,in how ways can the crew be arranged? Basic formulae of combinatorics(Permutations and combinations) Well I do not have good knowledge of boats, but I used the following...

Hey guys, I'm thinking of self studying some math this summer, and combinatorics is one field I never really got into but want to learn a bit of. I rigorous proof experience from reading other books/taking rigorous classes, so I wanted to start off with something relatively rigorous. Thanks

More a question to some conceptual understanding of combinatorics. The number of ways of picking Na elements to be in a box A, Nb to be in a box B and Nc to be in box C is given by: N!/(Na!Nb!Nc!) One can proof this by saying: Suppose we start by putting Na in box A and so on. Now I have...

I came across an interesting combinatorics optimisation description for 14numbers. Maybe someone good in combinatorics can expand this fano plane concept to a full 45 numbers lottery design. Any suggestions? http://en.wikipedia.org/wiki/Transylvania_lottery

The problem is : In how many ways can the letters of the word ASSASSINATION be arranged so that no two Ss are next to each other ? Tried taking the interval to the first S as x1, from then to the next S as x2...etc. with sum being 9. Possible partitions , including repeats, comes out as...

Homework Statement Prove that a! b! | (a+b)!. Homework Equations Probably some Number Theory Theorem I can't think of at the moment. The Attempt at a Solution Without loss of generality, let a < b. Therefore b! | \Pi _{k=1}^b a+k. Since (a+1) ... (a+b) are b consecutive...

Here's a very interesting question:- Suppose you are given a cube and 6 different colours. You are asked to colour each face of the cube with a different colour. If so, how many different colorings are possible? The thing is, this question looks easy to me at first look; the answer seems...

Hi everybody, I would really appreciate some help with the following problem. First of all I want to apologize for my poor English, I hope to be able to translate everything clearly. Thanks in advance. Homework Statement Find a number of 5 different digits that equals the sum of all...

I had a brain teaser that I was hoping you could solve: "In order to guarantee all emergency call are received, the phone company wants all phone nubers with 9 follower by 2 1's to be routed to 9-1-1 emergency center. This will ensure that any real emergency victims will still get help they...

Homework Statement I have a graph, with 4 vertices and 4 edges, i.e. a square. I need to find all non-isomorphic orientations of this graph. Homework Equations "Orientation of a graph arises by replacing each edge {x,y} by one of the arcs (x,y) or (y,x)." The Attempt at a Solution...

There's something I can not understand about proofs in combinatorics. Whenever I solve a counting problem, there's a non-negligible amount of uncertainty about the solution which I really don't feel when I solve problems in other fields, say in analysis or abstract algebra. It happens too often...

Homework Statement For n ≥ 1, let g(n) be the number of ways to write n as the sum of the integers in a sequence of any length, where each integer in the sequence is at least 2. For n≥3, show that g(n) = g(n-1) + g(n-2).2. The attempt at a solution I've gone through values of g(n) for...

Homework Statement How many non- empty subsets of {1,2,3...,15} have the following two properties? 1) No two consecutive integers belong to S. 2)If S contains K elements, then S contains no number less than K . Homework Equations choosing identities, not sure which one The...

Homework Statement A telephone extension has four digits, how many different extensions are there with no repeated digits, if the first digit cannot be zero? Homework Equations P(n,r)=n!/(n-r)! The Attempt at a Solution For the first digit, there are 9 possibilities (because no...

Homework Statement In how many ways can 24 cans of Fanta and 24 cans of Cola be distributed among five thirsty students so that each student may (a) at least two cans of each variety? (2p) (b) at least two cans of a variety, at least three cans of the other variety? (3p)Homework Equations The...

Homework Statement The screwengineer Pelle has a box with 16 black screws and 16 white screws. a. In how many ways can he pick an even(at least 2) amount of screws from the box? b. Pelle randomly chooses one of the alternatives in (a), what's the chance that gets as many white as black...

Homework Statement I am using a book that doesn't have any solutions in it, so I would like to be sure that I am doing the problems right before I move on. The question is below: In a programming language, a variable name must start with a letter or the underscore character, and...

Hi, I'm looking at the problem of counting lattice paths between two points with n steps, and I think I have my head around the case without boundaries to bump into (see e.g. http://www.robertdickau.com/lattices.html). However I'd like to work out the answer for the case of my path not being...

Please help me with combinatorics: I have three sets: A with elements (a1 ……..ai ) B with elements (b1 ……..bk ) C with elements (c1 ……..cj ) Pls help me to work out how many sets ABC shall I have that contain each element from sets A, B and C Thanks much

I'm a math major, and am planning on taking Applied Combinatorics next semester for an upper level requirement. I've looked it up, but still don't have a clear idea about what the course is about exactly. Also, is it considered a hard course?

Hi, I'm currently taking a Discrete Mathematics class and cannot seem to work out this one problem, we need to find the x^10 term in order to determine its coefficient of the equation f(x)=(x+x^2+x^3+x^4+x^5+x^6)^3 I know the answer is to be 27 from a previous problem (we are to use this method...

Homework Statement How many eight-letter passwords using the letters A-Z are there in which up to one letter is allowed to be used more than once? Homework Equations The Attempt at a Solution I broke the problem up based on repetition of one letter: (26)8 ways with no...

Homework Statement In a soccer tournament of 15 teams, the top three teams are awarded gold, silver, and bronze cups, and the last three teams are dropped to a lower league. We regard two outcomes of the tournament as the same if the teams that receive the gold, silver, and bronze cups...

Homework Statement My professor gave us this problem, I asked him about it but he just dumped it on me by saying "check it out yourself". Problem Determine the number of all combinations of winning numbers for Lotto 6/49 The Attempt at a Solution I read on wikia there are...

Homework Statement 45.) Twenty different books are to be put on five book shelves, each of which holds at least twenty books. a) How many different arrangements are there if you only care about the number of books on the shelves (and not which book is where)? b) How many different arrangements...

Homework Statement How many r-combinations are there of a multiset S = {1*a1, infinity*a2,...,infinity*ak}?Homework Equations The number of r-combinations on a multiset with k objects and infinite repetition number: (r+k-1) choose (r), or (r+k-1) choose (k-1). The number of r-combinations on a...

Homework Statement A teaching event takes two days and involves n people. Some of the people give a talk on day 1, some others give a talk on day 2. Everybody gives at most 1 talk, and there can be some teachers who do not give a talk in either of the two days. At the end of the event, a...

Homework Statement A president, treasurer, and secretary, all different, are to be chosen from among the 10 active members of a university club. How many different choices are possible if: a) There are no restrictions. b) A will serve only if she is the treasurer. c) B and C will not serve...

Hi everyone, I want to make sure if I solved this problem correctly. Thanks in advance. Homework Statement Rachel invited her friends to dinner. She has 10 friends, but only 6 places to sit them in her (circular) table. a) Count the ways to sit the guests if order is not important. b) If...

Homework Statement A circular table is arranged so as to have 9 different robots occupy the table. If there are 5 different types of robots, what is the number of possible arrangements of these robots? Homework Equations The Attempt at a Solution If it wasn't a circular table...

1. There are x distinguishable ants and there are x pots full of food. Due to the smell, the ants arrange themselves in a circle on the circular rim of each pot. In how many ways can they do this? note: Any number of pots can be free of ants. For example all the ants can be on the circular...

I am dealing with the following interesting combinatorics problem, several reformulations of which haven't help me solve the problem: Suppose we have a circular arrangement with M spots and we want to distribute N tokens over these spots such that there are token numbers n_m that respect...

Homework Statement I need to show that {^{2n}}C_n is an even number.Homework Equations {^n}C_k=\frac{n!}{k!(n-k)!}The Attempt at a Solution It was simple to convert the expression into factorials, but from there I really am stuck...

Homework Statement Let A be a 100x100 matrix such that each number from the set {1,2,...,100} appears exactly 100 times. Prove that there exists a row or column with at least 10 different numbers. Homework Equations The Attempt at a Solution I suspect that I should use the...

Hi. OK in a box there are 6 balls, 2 red ones and 4 blue ones. We take 2 balls out of the box without putting any of them back 1) If I wish to know the probability of selecting 2 blue ones, I just do this: (4above2)/(6above2)=6/15 or 4C2/6C2 or (4*3/2)/(6*5/2)2) BUT, if I wish to know the...