Counting Definition and 392 Threads

Suppose that for some application it is mathematically convenient to represent certain objects of interest (e.g., lines or conics) as n-dimensional vectors. That such a representation exists let's us conclude that in order to specify such an object, no more than n values are necessary. That is...

Homework Statement How many anagrams with 4 distinct letters and that have two of the letters "a", "b" and "c" can you make using the first ten letters of the alphabet? Homework Equations The Attempt at a Solution First I assume that by anagram they mean letters arranged in any...

Homework Statement You have 10 pieces of fruits of which 1 is an orange, 1 is an apple, 1 is a pear, and 1 is a peach. How many ways are there to organize them in a row if the apples and oranges must be together and the pears and peaches cannot be together. Homework Equations Mostly...

Hi everyone, I've hit a bit of a snag with part c of this problem (can't figure out how to invert a function T(ν)), so I'm starting to question whether I have the previous parts correct. Homework Statement Consider a system of N identical but distinguishable particles, each of which has a...

Problem: -The Union of set A, set B and set C has 104 elements. -The Union of Set A and B has 51 elements -The Union of Set A and C has 84 elements -The Union of Set B and C has 97 elements -The Intersection of Set A and the Union of Set B and C has 17 elements. -Set C has twice as many...

Hi, say I have a vector that is currently being populated with some objects. I would like, as each object is added to the vector, to count the maximum number of equal objects currently in the vector. Would you know of any efficient algorithm that does this? Thank you.

Homework Statement A detector is used to count the number of gamma rays from a radioactive source if the number of counts is 10000 in exactly 20 secs then what is the error in counting rate per sec? Homework Equations- no idea. The Attempt at a Solution- As per our syllabus we...

Hi. Recently I've been wondering if i could generalize the following problem I randomly had in mind: There are 15 kinds of identical books in a book store . The book store owner packs books into boxes that can contain up to 4 books . Assume that all books are of the same size and those boxes...

The electric field in a cubical cavity of side length L with perfectly conducting walls is E_x = E_1 cos(n_1 x \pi/L) sin(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t) E_y = E_2 sin(n_1 x \pi/L) cos(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t) E_z = E_3 sin(n_1 x \pi/L) sin(n_2 y \pi/L)...

Hi, For example, say we have the binary integer 101010 and I split it into blocks, say (1)(01)(01)(0). The size of the blocks is determined by some function. We want to count the number of '1' bits inside each of these blocks. I'm interested in a fast C++ code to do this. There are fast...

So over the course of yesterday and the day before that I've spent a few hours thinking about these problems; 1. Given a circle, place n points arbitrarily* on the edge and connect every point to every other point with a straight line. How many intersections do you get? 2. Given the same...

First I tried to make an organized list but I kept on messing up. Then I tried to subtract the cases that don't work but this also was not a good approach. Could I get some help on setting up the cases I should be considering?

Homework Statement When you shine a light on the ripples in a tank you get dark and bright patches like this: http://img687.imageshack.us/img687/4343/rippletank.jpg So my question is: how do we count the number of waves? In my experiment I am trying to find the wavelength by...

Homework Statement For this problem assume that the 365 dates of the year are equally likely as birthdays. b)Find the probability that in a group of n people, at least two have the same birthday. Homework Equations The Attempt at a Solution Well I am not really sure what...

I recently came across this theorem for the first time and read lot of theory however I am really confused with all the examples - specially 3d objects. So I decided to pick up simpler version. I have attached the diagram If a square is made up of eight equilateral triangles - using black...

Counting and Probability...why is this wrong?? Homework Statement How many ten letter words can be made from the english alphabet which contain at least one X, at least one Y and at least one Z? Homework Equations The Attempt at a Solution 1) Choose a place for X (10 ways)...

Dear forum members, I have a small problem counting all the invertible matrices of the size 2x2 providing \mathbb{Z}_{n}. This problem was difficult for me so I decided to go on counting how many invertible 2x2 matrices there are for n=32. My strategy to solve the problem was first by...

Homework Statement How many eight-digit numbers can be formed under each condition? a) The leading digit cannot be zero, the fifth digit cannot be 6 or 8, and the number must be less then 75,000,000. b) The leading digit cannot be zero, the number must be divisible by 5, the fourth digit...

Homework Statement There are two locks on the door and the keys are among the six different ones you carry. In a hurry, you dropped one somewhere. What is the probability that you can still open the door? What is the probability that the first two keys you try will open the door? The...

Homework Statement Gamelin VIII.1.6 (8.1.6) "For a fixed number a, find the number of solutions of z^5+2z^3-z^2+z=a satisfying Re z > 0" Homework Equations The argument principle relating the change in the argument to the number of zeros and poles of the function on the domain. The...

Homework Statement We draw 5 cards out of a normal deck of 52 cards. No replacements. How many combinations have exactly one Jack, one Queen, one King? Homework Equations The Attempt at a Solution The solution = 49,920 My attempts: I keep getting 99,840. So 4C1 Jack...

Homework Statement Determine the no. of energy levels with different values of total spin for a system of N particles with spin 1/2 The Attempt at a Solution Let f be the no. ways to get a z projection of spin, then f(\sigma )=Binomial(N,\frac{1}{2}N + \sigma) Where Binomial(a,b) is a chose b...

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i have a question about the relation between the riemann zeta function and the prime counting function . one starts with the formal definition of zeta : \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} then : ln(\zeta (s))= -\sum_{p}ln(1-p^{-s})=\sum_{p}\sum_{n=1}^{\infty}\frac{p^{-sn}}{n} using the...

greetings . i have a couple of questions about the prime counting function . when \pi _{0}(x) changes by 1, then it's logical to assume that it should happen at a prime argument . meaning : \lim_{\xi\rightarrow 0}\pi _{0}(x+\xi )-\pi _{0}(x-\xi )=1 implies that x is a prime . is this a true...

Homework Statement Count all surjections of A to B, (f: A ---> B) where |A| = |B| + 1 Homework Equations None? This is just a problem I came across online. The Attempt at a Solution I'm really not sure. This isn't technically homework, but I'm just looking for a good method. I...

Counting with composites... Please advise What is the standard for counting with composite numbers? If I can count to infinity with all odd composites and factor them, would that be considered key to counting all primes to infinity. By default all missing odd numbers in the sequence would be...

Hi all, i'm currently working on a project but i need to find a way to count bacteria i have to be able to tell the difference between 10^4 and 10^5 bacteria/mL. This should be done optically but I'm open for any suggestions. I have read something about optical density and flow cytometry...

This is a pretty basic counting problem, but it is confusing me to no end. I know the answer (from the back of the book), but I just don't understand the answer. Homework Statement Find the probability of getting exactly 4 numbers correct in a lottery where 6 numbers are chosen from 49...

I am struggling with convincing myself that if you equip \mathbb Z with the counting measure m, the L^p norm of measurable functions f: \mathbb Z \to \mathbb C looks like \| f \|_p = \left( \sum_{n = -\infty}^\infty |a_n|^p \right)^{1/p}. I know that any function on \mathbb Z is...

Hey Guys, So, I am trying to prove the Fundamental Counting Principle using induction. I have no clue where to start or even how to use induction to prove it. I would appreciate some help. The Question in a mathematical form: Let ^\sharp (A) = m and ^\sharp (B) = n. Proove by induction...

Homework Statement How many different ways can £2 be made using any number of coins? (In other words, how many ways can you obtain the sum of 200 with terms from the following finite set - 200, 100, 50, 20, 10, 5, 2, 1. Order does not matter.) Homework Equations None? The Attempt at...

Homework Statement 1+2+3...+n Homework Equations count the sum. The Attempt at a Solution it's such an easy sum to count but i just want to make sure. S_{n}=\frac{n}{2}(a_{1}+a_{n}) S_{n}=\frac{n}{2}(1+n) S_{n}=\frac{n+n^{2}}{2}

Question about shifting the indexes [nth powers] and counting numbers of series? When you multiply a series by x, therefore changing the nth power on x, is it mandatory that you change the number where you start counting? This is confusing me because some problems do both, where others explain...

Homework Statement I already have the solution to the problem. Just need some help deciphering the logic behind it. There are V balls, which are identical except for their color. N of them are blue and V-N are red. We place the balls inside a box so that V/2 balls are on each side...

Hi, I'm reading Becker2Schwarz, chapter 5.1, about D0 branes in the GS formalism. They introduce kappa-symmetry, and end the section with "without this symmetry there would be the wrong number of propagating degrees of freedom". I'm trying to understand that. The fermions \Theta^a have...

Homework Statement Question 1: a) Suppose you have brought four pens of different colours to the exam. For each of the ten question on the exam, you choose one pen. In how many ways can this be done? b) In how many ways can you distribute six bananas and five oranges between three children...

Hi All, I'm terribly stuck on this problem. We were asked to calculate how many hand gestures are possible, keeping in mind that a hand gesture consists of raising one or both hands and extending some fingers (note: raising just a fist is also considered a gesture). I started this problem by...

Homework Statement G is a group. H is a subgroup. lHl= order of H lGl=order of G [G:H]=Number of cosets Counting Formula lGl = lHl [G:H] I have a question of clarification about this formula. My book says that [G:H]=number of cosets. The problem is that at this point in my...

Homework Statement How many natural numbers are there with the property that they can be expressed as the sum of the cubes of two natural numbers in two different ways. Homework Equations N/A The Attempt at a Solution I don't understand how should i start. :( Can somebody give...

Counting Lists With Repetition Homework Statement How many ways can you create an 8 letter password using A - Z where at most 1 letter repeats? Homework Equations The Attempt at a Solution I'm not sure how to attack this problem but first I thought that A-Z considers 26 letters...

How many ways can you place 8 distinct flags on 3 distinct poles if no pole can be empty.Im not sure how to approach this problem because writing out all the possibilities would take a lot of time So I was thinking it would be something like 8C3 to select the three flags that have to be placed...

Homework Statement There are n distinct balls, and n distinct boxes, and one right order for the boxes to be in. What is the chance that none of the balls are in the correct box. And each ball can go into only one box. Homework Equations The chance that none of the boxes are in the correct, is...

Take 4 pennies and 2 dimes. Now assume that you have no way to distinguish the pennies from each other and no way to distinguish the dimes from each other, but you can tell the difference between a dime and a penny. For each of the following situations, how many different ways can you pull...

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...

What is the total number of bones in rabbit and frog? Can anyone help me to group them all...

Homework Statement "Find the probability of the sum of three rolled die being less than or equal to 9." Given problems like this, my question is: is there some way to find the answer besides writing out all 216 combinations and then counting the ones that equal 9? I feel like there is, but...

How come AI never makes an progress? OK this may be slightly over stated but it has been 50 years since we thought we could make an AI soon and we are no where near an AI.

Say you have a free particle, non relativistic, and you want to calculate the density of states (number of states with energy E-E+dE). In doing that, textbooks apply periodic boundary conditions (PBC) in a box of length L, and they get L to infinity, and in this way the states become countable...

Hi and thank you for reading this. I'm learning to use mathematica and among those things I'm trying to do, is to define a function that can count for me, say, the number of positive zeros less than a given number Z of a familly of function. For exemple, let f_n(x) = sin(x/n) for any...