# What is Counting: Definition and 412 Discussions

Counting is the process of determining the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.
Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...).
There is archaeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of social and economic data such as the number of group members, prey animals, property, or debts (that is, accountancy). Notched bones were also found in the Border Caves in South Africa that may suggest that the concept of counting was known to humans as far back as 44,000 BCE. The development of counting led to the development of mathematical notation, numeral systems, and writing.

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1. ### Basic counting methods — integers between 1 and 100 that are divisible by 7

There are 14 integers between 1 to 100 that are divisible by 7 If we used indirect approach, then we would do the total number of possibilities C(100,2) = 4,950 and subtract all nubmers that are indivisible by 7 : C(100-14, 2) = C(86,2) = 3,655 the difference of these two numbers is 4,950 -...
2. ### I What is the calculation for value B in Penrose's entropy model?

In the book "Cycles of Time" by Roger Penrose, there is a part of the explanation of entropy that I don't understand. There are 10^24 balls, half of which are red and the other half blue. The model is to arrange the balls in a cube with 10^8 balls on each edge. It also divides the cube into...

4. ### A Counting on a computing system

It is plretty clear that a classical computer can't count over the set of natural numbers. If it is a digital device and you used an infinite loop you would eventually run out of memory space and have to reinterpret the meaning of the numbers (so it isn't really counting independently). An...
5. ### Counting letters in a number between 1 and 1000

Excluding spaces and hyphens, how many letters are there in a given number between 1 and 1000 (inclusive)? How can I make my algorithm more efficient? dig_lens = [4,3,3,5,4,4,3,5,5,4] tens_lens = [0,3,6,6,6,5,5,7,6,6] teens_lens = [0,6,6,8,8,7,7,9,8,8] def num_letters(n): nlist = []...
6. ### Combinatorics: Counting quadrilaterals in triangle pattern

Hi, I was watching a Youtube on combinatorics (here) and a problem was posed at the end of the video about counting the number of quadrilaterals. Question: How many quadrilaterals are present in the following pattern? Attempt: The video started with the simpler problem of finding the number...
7. ### MHB Counting Color Combinations in 12 Triangles

There are 12 triangles (picture). We color each side of the triangle in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?
8. ### How to solve an unclear geometry problem — Counting line segments in a rectangle

120-48=72 72/4=18 The solution appears too simple to be the correct to me.
9. ### MHB Counting the elements in sets

Question: How many elements are in each set? For the first set, I think it's 8995 because the set is the union of {1,2,3,4,5},{1,2,3,4,5,6},...{1,2,3,...9000}. So 9000 - 5 = 8995. For the second set, I'm not too sure about counting the elements in the set. Since 1<x≤i, I can't think of any x...
10. ### Comp Sci Counting Valleys // Complete the countingValleys function below.

// Complete the countingValleys function below. // The code is in javaScript function countingValleys(n, s) { let currentLevel = [] // an array of numbers that indicate the number of units of altitude above or below sea level regarding each step let altitude = 0 // the...

12. ### Solving the Debate Over Photon Counting With a Scintillation Device

Hello, I'm working on a scintillation device to detect protons, I have a disagreement with one professor and I would like your opinion. There is one photodiode model we want to use to measure the light intensity from the scintillators, and we want to relate the signal of that photodiode with...
13. ### MHB Counting Problem: In a school 315 girls play at least one sports

In a school 315 girls play at least one sport. 100 play a fall sport, 150 play a winter sport, and 200 play a spring sport. If 75 girls play exactly 2 sports, how many play three?
14. ### I Counting Corners on a Moving Grid: Exploring a Fun Mathematical Problem

Summary: interesting counting problem for fun Imagine we draw a circle with diameter d and mark off sixty equal intervals like minutes on a clock. Then we draw two diameters perpendicular to one another and divide each in sixty equal intervals. Using the intervals on the diagonals we lay out a...
15. ### MHB Zero: Natural or Counting Number?

Zero is an integer. An integer is defined as all positive and negative whole numbers and zero. Zero is also a whole number, a rational number and a real number, but it is not typically considered a natural number, nor is it an irrational number. Why is 0 not a natural or counting number?
16. ### Troubleshooting 74LS90 Counting Issues: Using Input Pulses from a 555 Timer

I am trying to use the 74ls90. It won't change it's output. It suppose to count to 9. The input Pulses come from a 555 timer.
17. ### Counting Principles: 15 Ways Explained

I thought the number of ways would be dependant upon the number of toys. Since the number of toys isn't given I tried taking into the different ways you can order using different number of boxes. First situation: They can use all three boxes 3x2x1=6. Second situation: They can only use two...
18. ### Is the length of a day actually increasing by 0.21 seconds?

Since the length of day is greater by 0.21 seconds, thus the change in time = ##\frac{0.21*365*100}{60*60}## hours = 2.1 hours. Here, I do not understand why the length of the day increases by 0.21 seconds instead of 0.2 seconds.
19. ### Is my Proof Valid for Bijection of Finite Sets?

<Moderator's note: Moved from a technical forum.> Hi PF, I am learning how to prove things (I have minimal background in math). Would the following proof be considered valid and rigorous? If not any pointers or tips would be much appreciated! Problem: Prove that the notion of number of...
20. ### Counting the number of set-subset pairs within H

Homework Statement ##H=\{1,2,3,\dots , 10\}##. Determine the following number: ## \#([G,F];G\subseteq F \subseteq H)##. Homework EquationsThe Attempt at a Solution Here is my reasoning. If ##G=\emptyset##, there are ##2^{10}## ways to choose ##F##, so ##\binom{10}{0}2^{10}## total. If...
21. ### Counting a set in two ways #2

Homework Statement prove the summation by counting in a set two ways. Mod edit: The summation was later confirmed by the OP to be ##\sum_{k = 1}^n 2^{k - 1}## $$\sum_{n=0}^k 2^{k-1} = 2^{k}-1$$ Homework EquationsThe Attempt at a Solution [/B] ##2^{k-1} = (1+1)^{k-1} = \binom n 0 + \binom n 1...
22. ### Can the summation formula be proven by counting a set in two ways?

Mod note: Fixed numerous problems with LaTeX Also, the fractions shown throughout should instead be binomial coefficients 1. Homework Statement prove the summation formula by counting a set in two ways $$\sum_{k=0}^n k\left( \frac n k \right) = n *2^{n-1}$$ Homework Equations LHS = ##k...
23. ### I |Li(x) - pi(x)| goes to 0 under RH?

Extremely quick question: According to http://mathworld.wolfram.com/PrimeNumberTheorem.html, the Riemann Hypothesis is equivalent to |Li(x)-π(x)|≤ c(√x)*ln(x) for some constant c. Am I correct that then c goes to 0 as x goes to infinity? Does any expression exist (yet) for c? Thanks.
24. ### How Does Penrose Calculate Entropy in Cycles of Time?

Sir Roger Penrose in his book Cycles of Time on page 19 states the result of a calculation of probability of mixing red and blue balls as an illustration of entropy as state counting and the Second Law. He assumes an equal number of each. There is a cube of 10^8 balls on an edge subdivided into...
25. ### Counting problem - how many ways....

Homework Statement Pamela has 15 different books. In how many ways can she place her books on two shelves so that there is at least one book on each shelf. (consider the books in each arrangement to be stacked one next to the other, with the first book on each shelf at the left of the shelf)...
26. ### B NYTimes: 79 Moons of Jupiter and Counting

By Kenneth Chang July 18, 2018 On Tuesday, scientists led by Scott S. Sheppard of the Carnegie Institution for Science announced the discovery of a dozen moons around Jupiter, bringing the total number orbiting the solar system’s largest planet to 79. Next to the famous moons that Galileo...
27. ### Determining the time interval in a radiation counting problem

Homework Statement Given: Cn_dot = true event rate = 10 interactions/s p(t')dt' = differential probability of an event Homework Equations p(t')dt' = Cn_dot * exp(-Cn_dot * t') dt' The Attempt at a Solution [/B] I want to sample the time interval using python. But I'm not sure how to go...
28. ### How many number of IP addresses are available?

Homework Statement Counting Internet Addresses In the Internet, which is made up of interconnected physical networks of computers, each computer (or more precisely, each network connection of a computer) is assigned an Internet address. In Version 4 of the Internet Protocol (IPv4), now in use...
29. ### B Problem in Counting - Number of Passwords

Problem: How many passwords can be created with 6 to 8 characters. Letter case does not matter. Every password must have at least 1 digit. Approach taken in the solution in the book: Passwords with 6 characters P6 = 36^6 − 26^6 = 1,867,866,560. Similarly, we have P7 = 36^7 − 26^7 =...
30. ### Counting clicks (signals) at a radio frequency receoved by a shortwave radio

Hello. I'm an English language teacher researching the details of a class project; I use projects to develop language skills. I have a background in mechanical engineering but know little to nothing of electrical engineering, hence this post. Questions for members of this forum: Electrical...

33. ### Counting strings of alphabets 0,1,2 and 3

Homework Statement Consider the collection of all strings of length 10 made up from the alphabet 0, 1, 2 and 3. How many of these strings have weight 3? How many have weight 4? How many have even weight ? Homework Equations Combinations formulae The Attempt at a Solution Let me explain what...
34. ### B Counting intersections of lines in a triangle

As a result of working on https://www.physicsforums.com/threads/area-of-hexagon-geometry-challenge.914759, this question occurred to me: Divide each side of a triangle into n equal lengths. Connect the ends of each length to the opposite vertex with straight lines, thereby forming 3n...
35. ### How do you represent three with your fingers?

Okay, so my wife and I just had a huge arguement. My wife and I were positive that everyone did it that same way that we did. So what do you do. Me: 1- index, 2- index/middle, 3- index/middle/thumb, 4- all fingers My wife never puts her thumb up until five. Then someone started with their...
36. ### I Dimension using box counting technique

On an exam we just took, we were asked to find the dimension of a set using the box counting technique. So choose an epsilon, and cover your object in boxes of side length epsilon, and count the minimum number of boxes required to cover the object. Then use a smaller epsilon and and count the...
37. ### Counting operators with group theory

Homework Statement I have an exercise that I do not know how to solve. ##N## is a nucleon field, in the fundamental representation of ##SU(4)##. We want to classify operators by their ##SU(4)## transformation properties, bearing in mind that the nucleon is a fermion and we need antisymmetric...
38. ### Aromaticity of Conjugated Ring - Counting Pi Electrons

My professor drew the following molecule on the board and asked us how many pi electrons this aromatic molecule has. Everyone in the class said 14, as there are 7 double bonds, with two pi electrons from each bond. He told us that there are only 10 pi electrons in this molecule, refused...
39. ### Engineering JK Flip Flop Not Counting Properly in Actual Circuit

Homework Statement Hello everyone. I tried practicing creating up down counters using the J/K FF. I've been using the Synchronized Clock with an up down counter and the simulation seems to be successful based on this circuit on the image However, in the actual circuit in a breadboard using a...
40. ### I Semiclassical state counting

Long time ago I encountered a claim that if you fix some energy interval [E_A,E_B], the measure of the set \{(x,p)\;|\;E_A\leq H(x,p)\leq E_B\} where H(x,p) is some classical Hamiltonian, is going to be approximately proportional to the number of energy eigenstates contained in the energy...
41. ### Counting degrees of freedom in field theory

I'm having a bit of trouble with counting the number of physical ("propagating") degrees of freedom (dof) in field theories. In particular I've been looking at general relativity (GR) and classical electromagnetism (EM). Starting with EM: Naively, given the 4-potential ##A^{\mu}## has four...
42. ### I Calculating Modes in a Cavity: Why Use a Spherical Volume?

My question stems form the section "How Many Modes in a Cavity?" in the following derivation of Rayleigh-Jean Law: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html#c2" In here, they count the number of modes as represented by volume of an eighth of a sphere. What's the mathematical...
43. ### Counting Lines in a File: A Homework Statement

Homework Statement #include <stdio.h> #include <stdbool.h> #define MAX_STRING 256 #define MAX_LINHA 1024 typedef struct{ int dia, mes, ano; }Data; int contar_linhas(char fname[]) { FILE *f; Data int soma=0; // Inicializa o contador f =...
44. ### Counting elements of a field and more

Homework Statement Consider the field K=Z_2[X]/<x^3+x+1>. How many elements does this field have? Then let f(x) = x^3+x+1 and b be a root of f(x) in Z_2[X], Find all other roots of f(x) in K and describe the Galois group Homework EquationsThe Attempt at a Solution So I'm trying to understand...
45. ### A Counting Radial Nodes between Orbitals with Same l

Dear Everyone, Could anyone explain why we count only the number of radial nodes between the subshells that have the same orbital angular momentum l ? For example, 3p-orbitals have 1 radial node that exists between the 3p- and 2p-orbitals. Shouldn't be there additional radial nodes that exist...
46. ### Counting number of consecutive elt's in nit string

Homework Statement How many strings of length 10 contain either five consecutive 0s or five consecutive 1s? Homework EquationsThe Attempt at a Solution So this problem isn't too bad, and I have a pretty good idea of how to solve it. However, I am not sure whether the problem is referring to...
47. ### Prove H Contains in gH ≠ g-H: Counting Principles

Homework Statement Suppose that H is a subgroup of G such that whenever H a is not equal to H b. Then a H not equal to b H. Prove that g H g ^-H Contains H. For all g Is an element of G.[/B]Homework EquationsThe Attempt at a Solution I tried the contrapositive position ( sorry...
48. ### Understanding Combinations with Replacement in Probability

1. Homework Statement If a total of 5 distinct awards are distributed among 30 students where any student can receive more than 1 award, how many possible outcomes are there? 2. Homework Equations \text{outcomes} = r^n where r is the number of choices and n is the number of draws. 3. The...
49. ### Why doesn't (8c1)(13c2) work for choosing a team with at least one woman?

Homework Statement A club has only 8 women and 6 men as members. A team of 3 is to be chosen to represent the club. In how many ways can this be done if there is to be at least one woman on the team. Homework EquationsThe Attempt at a Solution I can do this 2 ways, first 1w2m + 2w1m + 3m0m...
50. ### Stargazing GAIA - 1 billion stars and counting

The GAIA telescope has been mapping stars in the Milky Way with unprecedented quality and quantities. It has been assembling the most detailed 3D map ever made of our Milky Way galaxy and has currently mapped over 1 billion stars. There are already hints that the Milky Way may be shaped...