Combinatorics Definition and 421 Threads

Hello Everyone, There is one interesting exercise in which it is asked to solve the following problem: In how many ways can we distribute 20 candies among 6 children so that the youngest gets at most 2 candies? This is my version of the solution: Case 1: youngest child gets no...

Homework Statement Suppose we put N atoms of argon into a container of volume V at temperature T. Of these N atoms, Nad stick to the surface, while the remainder Ngas = N - Nad form an ideal gas inside the container. Assume that the atoms on the surface are not able to move and have an...

I guess more and more people in the world tend to use virtual money (credit cards, direct debit cards...) to pay for their shopping. Here in Europe, however, people often still use cash in many situations. This obviously has a side effect concerning coins. Except when you go to a bar...

I want to learn combinatorics. Please send links? If possible, can you explain now?

Is this the right place to post a combinatorics question?

Homework Statement You have got 10 multiple choice questions with 3 possible choices per question. What is ur chance that you will have 5 answers correct? So the chance for a correct answer per question is 1/3 and for an uncorrect answer per question is 2/3. The Attempt at a Solution...

Hi everyone, I'm going to be taking a combinatorics course next fall, and I'd like to prepare over the summer for it. Here is a link to the class log from a few years back. I am currently in a graph theory course, but I've heard the combinatorics course is rather difficult. Does anyone...

(This is not HW, even though it may look a bit like it.) Using the notation nCr in the combinatorics meaning, The sum of ((n+k)Cn)*((1/2)^(n+k)) from k = 0 to n equals one. Why? (I thought it might use the identity (n+1)Cr = nCr+nC(r-1), but that didn't get me anywhere.) Thanks in advance for...

Hi all I need some assistance 1. Homework Statement with the attempt How many 5-digit briefcase combinations contain 1. Two pairs of distinct digits and 1 other distinct digit. (e.g 12215) I wasn't sure on which approach was correct. 10 * 9 * 8 (because there are three distinct...

Homework Statement I am trying to understand the question: An urn contains n red and m blue balls. They are withdrawn one at a time until a total of r(r≤n) red balls have been withdrawn. Find the probability that a total of k balls are withdrawn. The solution is given as, Sample...

Homework Statement n dices are thrown. In how many ways can you get an m (<=n) amount of fives? The Attempt at a Solution Well, I thought since you can organize the 5s in m! ways, and for each such permutation you can organize the remaining dice rolls in ##5^{n-m}## ways, the solution must...

Homework Statement What is the smallest number of coins needed to pay in exact change an change less then a dollar (coins are 1, 5, 10, 25 and 50 cents)?Homework Equations No relevant equations apart from a theorem: Numbers between n and k inclusive, k>n, if k - n + 1 The Attempt at a Solution...

Find the number of ways to arrange three types of flags on an n foot flag pole: red flags (1 foot high), white flags (1 foot high), blue flags (3 feet high) Find a recurrence relation for this number with one condition that there cannot be three 1 foot flags in a row (regardless of their...

In codes of length five over Z3 with d = 3 By using Hamming bound, the upper bound for the number of codeword is 22 and by Gilbert-Varshamov bound, the lower bound for the number of codeword is 2. The question is to show that you can do better than the lower bound from above by constructing a...

Consider S ={0120, 1010, 2011} as a subset of codes of length four over Z3 with d = 3 By (a) Show that S is a linearly independent set. I am asked to show S is a linearly independent set. However, if I add 0120 + 0120, I get 0210. Since 0210 is not in the set S, is S still a linearly...

I find myself too weak at this subject. Even spending a lot of time on problems doesn't give good results. I can do the easier problems and sometimes a few intermediate ones but everyone knows that these kind of problems aren't really asked in the exams. Anyone got some nice suggestions?

Given 5 grades how many different ways are there to assign them to 7 people? How do you determine that this should be 5^{7} and not 7^{5}?

1) an = an-1 + 2n with a0 = 2 The question is Use back-substitution to find a closed formula for the recurrence relations. I have an-4 + 2(n-3)+2(n-2)+2(n-1)+2n Then I have a1 + ... +2(n-5)+2(n-4)+2(n-3)+2(n-2)+2(n-1)+2n First of all, is my way right? If so, could you please help me find...

Homework Statement Here is some basic combinatorics, I need someone to check it for me, please before the lecturer :) Sorry for the stupid questions, hope I've made myself clear with the explanations of the answers given. (1)(a) If 8 cooks are to be divided among 4 restaurants, how...

Homework Statement 12 subjects (6 male, 6 female) are put on 3 lines, 4 in each. In how many ways can this be done, if one of the males and one of the females want to be on the same line? Homework Equations None. The Attempt at a Solution I thought it like this. I can pick the...

Let's say you have a group of 22 people, which you would like to break into 5 different groups -- 3 groups of 4 and 2 groups of 5. How many distinct ways can you form such groups? I don't want to double count groups. Let's say I number the people from A - V. The group ABCDE and ACDBE should...

http://i.imgur.com/O7iWyCF.jpg the table on this image shows a system of two einstein solids isolated from the environment. with three oscillators and a total of 6 units of energies (hf). can someone explain to me how they got they're ΩA and ΩB values?

For $n=1,2,...,$ set S_n=\sum_{k=0}^{3n} {3n\choose k} and T_n=\sum_{k=0}^{n} {3n\choose 3k}. Prove that $|S_n-3T_n|=2$.

Homework Statement I need to create a draw algorithm similar to the UEFA Champions League / Europa League draw. Example: I have 32 teams, up to 5 teams can be from the same country. I need to create: A) pairs (round robin style), no 2 teams from the same country can meet B)...

Find the number of different combinations of r natural.numbers that add upto n I tried this for quite a fair amount.of.time but.couldn't figure it out.(Punch)

Hi everyone, I hope your summer is going well. My name is Quinn and I am currently a junior CS and math major at VT. This summer I am doing some brain simulation research and I have come across a Combinatorics arrangement problem that I am stuck on. I was hoping someone here could shed...

Homework Statement Peter and Simon shares a bag of 11 fruits, of which 3 are poisoned. Peter eats 4 fruits, Simon eats 3 and their dog eats 1 fruit. What is the probability that both Peter and Simon gets poisoned, given that the dog ate a healthy fruit? Homework Equations The...

Homework Statement Let X be a set containing all four digit numbers made up of {1,2,3}, where every number contains every digit at lease once. Number of all subsets is: The Attempt at a Solution So firs i have to find number of elements in the set: 3!*3 + 3*12 = 54 Now what they...

I'm just whining about how every technical interviewer seems to think that "Oooh, I'll ask a complicated combinatorics question. That'll get them." Some one needs to explain to interviewers that combinatorics is another field that can be learned, and while it's more relevant than asking...

Hello MHB, I got difficult understand this kind of questions, anyone got any tips or could explain, cause I find it really hard to understand when I do this kind of problem, this is an old problem from exam that I hopefully did translate well. A priest needs to choose six hymns for Sunday's...

Question: Given a non-negative integer N, show many sets of non-negative integers (a,b,c,d) satisfy 2a+b+c+d=N Proposed (and roadblocked) solution: Case 1: 2a=0 Then there are \binom{N+2}{2} solutions (easy to prove). Case 2: 2a=2 Then there are \binom{N+2-2}{2} solutions. Case 3: 2a=4...

Homework Statement A small commuter plane has 30 seats. The probability that any particular passenger will not show up for a flight is 0.10, independent of other passengers. The airline sells 32 tickets for the flight. Calculate the probability that more passengers show up for the flight...

Hi, I wish to confirm the results I obtained for the two following questions in statistics. I'd truly appreciate your feedback. Homework Statement 1) Die 1 and die 2 form a pair of unbiased dice. Die 1 has 4 faces painted red and 2 painted blue, whereas die 2 has 4 faces painted blue and 2...

Homework Statement There are 7 different routes from A to B, 4 different routes from B to C, and two different routes from C to A. What are the possible routes from A to C and back, allowing any route to be traversed once in each direction.Homework Equations The Attempt at a Solution So I think...

Homework Statement Homework Equations {a \choose b} = \frac{a!}{(a-b)!b!}The Attempt at a Solution So I tried doing a mathematical induction proof (Show that it is true for some number, then show that if you assume it's true for 'k', it implies 'k+1') I was able to get the initial step, to...

When rolling 6 6-sided dice, how many different ways can you have exactly 4 different numbers? I tried solving this like so, the first dice has a possible 6 numbers, the second has a possible 5, the third has a possible 4, and the fourth, 3. Then there are 2 remaining dice of which each has...

Homework Statement An experiment has 3 outcomes. If the experiment is performed 4 times, how many sets of outcomes exist in which exactly two of the experiments had the same outcome. The Attempt at a Solution From what I understand there is a 4-element set with each element having 3...

Homework Statement There is a group of 7 people. How many groups of 3 people can be made from the 7 when 2 of the people refuse to be in the same group? Homework Equations nCr The Attempt at a Solution Here is what I know: 7C3 gives the total number of groups that can be...

Hi all, Can anyone gimme any clues to solve the sum below (or solve it outright :D)? \sum_{i=k}^{n} \frac{i!}{(i-k)!} I'm trying to solve one of Feyman's logic problems (bored geek alert) and I'm stuck at this point. And since my high school days are so far behind... Thanks in...

1. So consider an 8 megapixel picture (res: 3264x2448). Now, it seems rather simple but I just can't figure out how to calculate the entire number of possible shots/photographs one can take within that resolution, assuming each pixel can have 16777216 different values/colors. Homework...

Hello there, So consider an 8 megapixel picture (res: 3264x2448). Now, it seems rather simple but I just can't figure out how to calculate the entire number of possible shots/photographs one can take within that resolution, assuming each pixel can have 16777216 different values/colors.

Homework Statement The problem I am working on is: An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession. a.How many different possible PINs are there if there are no restrictions on the choice of digits? b.According to a...

Homework Statement How many "words" with 5 letters can be created from the letters in the word ALGEBRA? Each letter can be used only once. Homework Equations The Attempt at a Solution I know the answer to this (1320) and I know how I got to it (which I'll describe in a minute)...

Fifteen cities are planned to be connected in such a way that each city has precisely one road leading to each of five other cities.How many such roads are to be constructed? This question was asked in a talent test conducted in our school.

So I found a formula for the number of ways of coloring a shape with 20 triangular faces, 30 edges, and 12 vertices: (1/60)*(k^20+15*k^10+20*k^8+24*k^4). Now I need to find the # of ways of coloring the faces with exactly 5 colors each with each color used exactly 4 times. I know how to find...

I'm just checking my work on this. Given an 8x8 chessboard, you randomly place 8 rooks on the board. What is the probability that no rooks can capture another one. In other words, probability that no 2 rooks are in the same row or column. My solution is simply 8!/(64 choose 8), but that...

I have that B(x)=e^{e^{x}-1} is the generating function for the number of set partitions. Also the Stirling numbers of the second kind are defined by S(0,0)=1, S(n,0)=S(0,n)=0 for n=>1 and S(n,k)=S(n-1, k-1) + kS(n-1, k). Show that e^{u(e^{x}-1)}=1+\sum_{n\geq...

Hello MHB. I want to read an advanced combinatorics book. I am well versed in combinatorial methods like counting proofs, Pigeon Hole Principle, criticality, induction, graphs etc. I have good knowledge of undergraduate algebra so I am fine if the book you recommend has some algebraic methods...

So the question was, Let x > 0. Prove that x^{n} = \sum \frac{x!}{x!-k!} S(n, k). Where the sum goes from k = 1 to n and S(n, k) is th Stirling numbers. I believe I have proven what I needed to, but my question is why does x have to be greater than 0? Couldn't we define a function that maps...

6 friends go to a party, each one carrying a different umbrella. They place the umbrellas outside. When the party is over, they are drunk and each one grabs an umbrella at random. In how many ways could none of them have taken the right umbrella? I'm having a bit trouble with this, as I...