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. J

    Counting the possible number of shapes.

    Hello, I have an 8 by 8 binary matrix. Define a shape as a cluster of 1's. For instance, consider the following sample: 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 As you can see, there...
  2. D

    How Many 5-Character Strings Include At Least One '@' Symbol?

    Homework Statement Given that the ASCII character system has 128 possible characters how many 5 character strings are there with at least one occurence of the '@' symbol. Homework Equations The Attempt at a Solution So clearly which symbol we're using doesn't matter, and I see...
  3. C

    Lebesgue measure vs. box counting dimension

    Short version: What is the difference between the Lebesgue measure and the box counting dimension of a set? Long version: I was reading up on the definition of the Lebesgue measure, and the description of how to take the Lebesgue measure of a set (which I understood basically as "cover the...
  4. C

    What techniques can speed up counting pseudoprimes?

    I'm calculating the high-water marks for the following function: f(n) = #{n is a k-strong pseudoprime, 1 < k < n} where n is a composite integer. The naive Pari code: ff(n)=sum(k=2,n-1,isSPRP(n, k)) record=0;forstep(n=3, 1e6, 2, if(isprime(n), next); k = ff(n); if (k > record, record = k...
  5. S

    Counting cycles in a permutation

    I'm trying to show that for two permutations f ang g in Sn, the number of disjoint cycles in fg is the same as the number of disjoint cycles in gf. I know that in general fg does not equal gf, but by working examples it seems like they always decompose into the same number of disjoint cycles...
  6. J

    HTML - counting the number of responses from a text file

    HTML -- counting the number of responses from a text file Hello, I have an angelfire website and on one of my pages I ask a simple question. What is your shirt size? There are four options: Small, Medium, Large and Extra Large. The answer is then submitted to a text file. My problem is...
  7. H

    Canberra Multiport II Ginie 2000 Stops counting

    We have Canberra Multiport II with 5 ADCs and coupled to Ginie 2000 software using USB key, we are using only one ADC no #3 currently. Counting is carried out for preset number of counts under single ROI. The counting automatically stops mostly at around 2200 hrs or 0000 hrs or 1000 hrs. This...
  8. E

    Counting Number of Irreducibles

    Homework Statement Let p be a prime. (a) Determine the number of irreducible polynomials over Zp of the form x2 + ax + b. (b) Determine the number of irreducible quadratic polynomials over Zp. The attempt at a solution A nonzero, nonunit polynomial f(x) in Zp[x] is irreducible if it...
  9. M

    Counting Raindrops in Wichita, KS

    moe in wichita ks how many drops are in 1 inch of rain on a 1 inch dia tube? of course iam guessing that all rain drops are the same size, which most likely they are not. sounds the like this is going to have a lot averages in the answer.
  10. S

    Counting Problem: Inviting Jack's Friends to Dinner

    Homework Statement a) if jack has 10 friends, in how many ways can he invite 5 of them to dinner. b) suppose 2 friends don't like each other, and if one is invited, the other can't come. c) what if 2 of the friends are married and if they invite that friend, the spouse must come. Homework...
  11. C

    How many exchanges are needed to serve a city of 80,000 people?

    A seven-digit phone number in the United States consists of a three-digit exchange followed by a four-digit number. How many exchanges are needed to serve a city of 80,000 people? Combination, Permutation, and arrangement with repetition equations are used in this section. Part 1 of...
  12. rocomath

    How Many Sets of Four Consecutive Integers Have a Product Under 100,000?

    How many sets of four consecutive integers are there such that the product of the four integers is less than 100,000? Set_1=1,2,3,4 Set_2=5,6,7,8 Set_3=9,10,11,12 Set_n=a\cdot b\cdot c\cdot d<100,000 Okay, I know I could continue with my Sets, but there has got to be a more logical approach...
  13. P

    Path Counting - Chances of two people meeting?

    Homework Statement I am having trouble with this problem. A network of city streets forms square bloacks as shown in the diagram below. http://img182.imageshack.us/my.php?image=librarypoolqs6.jpg Jeanine leaves the library and walks toward the pool at the same time as Miguel leaves the pools...
  14. M

    How can permutations help with counting using probability?

    Hi All, I am trying to reconcile two approaches used in counting problems. The first approach uses combinations and the other uses probability. I understand the combinations approach, but not able to comprehend the probability approach. Consider the following example, A carton contains...
  15. M

    Using Sylow's Counting to Classify Groups of Order 44

    Hi last one here. Any hints on this is appreciated too :) Let G be a group of order 44. Show using Sylow's counting that G has a normal subgroup of order 11. Use the results to classify all groups of order 44.
  16. M

    What are the possible forms and classifications of groups of order 12?

    Hey there guys. Let G be a group of order 12. Show by a Sylow counting argument that if G does not have a normal subgroup of order 3 then it must have a normal subgroup of order 4. Deduce that G has one of the following forms: (i) C_3 \rtimes C_4 (ii) C_3 \rtimes (C_2 \times C_2) (iii) C_4...
  17. M

    Did I do this counting problem correct?

    Homework Statement The question says: A chain of stereo stores is offering a special price on a complete set of components (reciever, compact disc player, speakers). A purchaser is offered a chocie of manufactuer for each component: Reciever: Kenwood, Onkyo, Pioneer, Sony, Yamaha Compact...
  18. K

    Counting partitions with two givens

    Hi Is there a relatively easy way to calculate the number of partitions of a number given the maximum term and the count of terms? A couple of examples: 25 has four partitions with five terms where each term is unique and the largest term is 8 {8,6,5,4,2} {8,7,5,3,2} {8,7,5,4,1} {8,7,6,3,1}...
  19. T

    Pigeonhole principle and counting

    Homework Statement 1) 100 of the 5-element subsets of {1, . . . , Y } have the same SUM. (Fill in Y . Make Y as small as you can, however you need NOT prove that it is smallest possible. You might need a calculator.) 2) Let FUNC be the set of all FUNCTIONS from N to N. Show that FUNC is...
  20. D

    Probability; counting questions

    Let me prefix by stating that this is not really a homework problem, just something I am curious about. I posted it here because it is probably too easy to go into the "big boy" forums :smile:. Last night my wife said she needed a 23% on her final to maintain a grade of A in her course. I...
  21. Loren Booda

    Better than counting on fingers

    Consider line segments oriented vertically, horizontally, and either diagonally. The individual segments may be seen as stretching between an imaginary 12 and 6 o'clock; or 1:30 and 7:30; or 3 and 9; or 4:30 and 10:30. Thus give them singly or doubly a common center. By using them so -...
  22. E

    Counting Seating Arrangments of Couples at a Round Table

    [SOLVED] Counting Seating Arrangments of Couples at a Round Table I'm reading this example in my probability book which is I'm not understanding. It says: There are 19! ways of arranging 20 people around a table. The number of arrangements that result in a specified set of n men sitting next...
  23. C

    Math software that allows for working in different counting schemes?

    for example...is there some software that will let me do all my calculations mod 2 or something? free software would be preferable thank you
  24. C

    What's the effect of using modular counting in matrices?

    let's say for example, I am interested in using mod 2 integers ({0,1}) to get rid of certain coefficients. Now, I am most interested in eigenvalues. How will this affect my eigenvalues compared to the original matrix (normal counting)? Is there anyway I can "retrieve" the original eigenvalues?
  25. L

    Solving the Odd 3-Digit Number Permutations

    Homework Statement How many 3 digit numbers can be constructed from digits 1, 2, 3, 4, 5, 6, and 7 if each digit may be used once only and the number is odd? 2. The attempt at a solution What number do they speak of? The resulting 3 digit number? How do I approach this equation?
  26. E

    How Many Investment Strategies Exist for $20,000 Across 4 Opportunities?

    [SOLVED] Funky Counting Question Problem. We have 20 thousand dollars that must be invested among 4 possible opportunities. Each investment must be integral in units of 1 thousand dollars, and there are minimal investments that need to be made if one is to invest in these opportunities. The...
  27. W

    Beta particle counting efficiency?

    Beta particle counting efficiency?? Hi! Can anyone tell me what counting efficiency means? For example if it's put into a question as: "shielded beta counter with 85.7% counting efficiency, 845 counts are accumulated in one week" what does it mean? How I understand it so far is that it's...
  28. C

    Fun with counting and modular arithmetic

    So today I was doing a problem out of my book for practice, and I came across some interesting results. Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by 4. a set of consecutive integers 1 mod 4 = 1 2 mod 4...
  29. F

    Preparing for a Physics Exam: Two Weeks and Counting

    I am in my final year at high school and i have choosed physics exam, there left only two weeks until it, have anyone ideas how to prepare for it in such a short time?
  30. D

    How to Solve a Faded Safe Code: Math Counting Problem Explained"

    To open a safe, 4 number buttons must be pressed, in the correct order. Over time, the 4 numbers buttons of the code fade. A thief notices the faded buttons, so knows that the code consists of those 4 numbers. How many possible codes are there? 4 numbers can be arranged in 4! different...
  31. C

    Failure of the sylow counting argument

    show that there is a normal subgroup of G of order 5 when G is a group of order 30. My friend just called me with this problem, he said the usual method of solution fails. (i.e. using sylow and then showing that the subgroup is unique and deducing that it must therefore be normal), I told him to...
  32. R

    Why Can't I Sleep? Adventures in Insomnia

    Still awake, and not even drowsy at the moment! Maybe it's the cumulative lethal dose of caffeine, or the painful frostbite on my ear (-8C outside, very windy), or the pleasant warm glow of the gray PF background - I'm still going after 90,000+ seconds of wakefulness! (okay, excluding the brief...
  33. H

    Solve Counting Problem: How Many Students Left Unfinished?

    Homework Statement There are 285 math students. First homework was completed by 166 students, second by 148 and third by 129. First and second was completed by 108 students, first and third by 83 and second and third by 25 students. How many students have not completed at least one homework...
  34. C

    Counting solutions to the EFE?

    Way back in July 2004, kurious asked: Just thought I'd mention that Einstein himself proposed an interesting method for "counting" the solutions of a PDE which may have been indirectly inspired by the landmark work by his colleague David Hilbert on what is now called "the hilbert polynomial"...
  35. M

    Counting Problem: How to Get the Right Answer?

    Suppose you pick a k-element subset of {1, 2, ..., n}, call it A. How many of the other k-element subsets have k-1 elements in common with A? I've been at this for quite some time, but I always overcount. Can anyone help me out? My last attempt gave me (n-k+1)k - \frac{k(k-1)}{2}, which isn't...
  36. M

    Am i understanidng this right, counting strings

    Hello everyone. I'm not sure if I'm doing this right or not. The problem asks: Consdier the set of all strings of a's, b's, and c's. a. make a list of all of these strings of lengths zero, one, two, and three that do not contain the pattern aa. Okay so i have the following: note e: stands...
  37. M

    Counting problem, exactly 5 heads obtained, coin

    Hello everyone, I'm having some issues on this problem: A coin is tossed ten times. In each case the outcome H (for heads) or T (for tails) is recorded. (One possible outcome of the ten tossings is denoted THHTTTHTTH.) I got a, d right i believe. but I need someone to check if i did the...
  38. L

    Single photon counting experiences

    Hello, Single photons counting is daily life since decades in nuclear physics: gamma rays and X-rays are detected individually since very long. Solid-state detectors are the most used. In the visible part of the spectrum, this is becoming daily life also. Detectors have reached very high...
  39. A

    Theorem 10: Prime Counting Function and Loglog x

    I am going through Hardy's book on number theory.The following theorem I do not understand. theorem 10: pi[x] >= loglog x where pi[x] is the prime counting function and >= stands for greater than or equal to The arguments written in the book are very compact.please help .
  40. S

    Riemann Prime Counting Function

    Can anyone tell me if Riemann's Prime Counting function can be solved by residue integration? Here it is: J(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{ln(\zeta(s))x^s}{s}ds which has the solution: J(x)=li(x)-\sum_{\rho}li(x^\rho)-ln(2)+ \int_x^{\infty}\frac{dt}{t(t^2-1)ln(t)} I...
  41. N

    Counting Time: Can Someone Help?

    I have very disappointed myself. I am forgot how count time, if car drives example 45 km(kilometre) distance and it's speed is 100km/h ( kilometre per hour). How i know what is time? equation is t = s/v (t = time, s = distance and v = speed) and now equation is t = 45km...
  42. Mallignamius

    Does nature have a way of counting?

    What's the closest way that nature counts or does math? I was thinking either symmetry (perhaps crystal growth or structure) or population (maybe by equilibrium?), but that just doesn't feel complete. I guess those could be considered to be numbers, but there isn't any computation going on...
  43. S

    Is Every Set Containing a Countably Infinite Subset Uncountable?

    This is the question, and we're supposed to answer if it's true or false: If A is a countably infinite set, and A is a proper subset of another set B, then B is uncountable. I thought this was false, because if A is infinite and countable, then B should also be infinite and countable in...
  44. G

    C/C++ Counting Integers: Writing a Program

    How I can write a program which reads a sequence of integers and counts how many there are. Print the count. For example, with input 55 25 1004 4 -6 55 0 55, the output should be 8 because there were eight numbers in the input stream. Please help.
  45. C

    Why Is My C Program Not Counting Vowels Correctly?

    Hi guys! I made a program in C, which count vowels on text, but it doesn't work. Thats the code i have writed: #include <stdio.h> #include <string.h> int main() { char buffer[80]; int counter; printf("Enter a line of text: "); fgets(buffer, sizeof(buffer), stdin)...
  46. M

    Are These Solutions to Counting Problems Correct?

    Hi, I've done more questions. Hopefully the wordings on these word problems aren't vague. Could someone take a look? I'm not sure if I've thought about the problem the right way. :smile: Thanks v. much in advance.
  47. M

    Counting question - check reasoning

    Hi All, I'm taking discrete math as part of my computer science course. i don't quite understand why i didn't get the answer in the book. Please take a look at my attachment and see where have i gone wrong. Thanks. :blushing:
  48. C

    Ordering Dozen Doughnuts: How Many Ways?

    The doughnut shop has 5 kinds of doughnuts: a, b, c,d and e. There are unlimited supply of each kind. In how many ways can you order a dozen doughnuts? Well, my first instict is to simply 5^12. But then I realize aaaab is the same as baaaa... hence the order doesn't matter. I'm trying...
  49. N

    Counting Divisors of 388,800 | Even and Odd Factors Explained

    1.The integer 388,800 can be factored with primes as 2^6 × 3^5 × 5^2 (a) How many unique divisors does 388,800 have? (b) How many of these factors are even? odd? I have no clue how to do this. There are no similar examples in our textbooks or notes. I searched up counting divisors on the...
  50. E

    Counting Particles Passing Through a Hole

    I want to find out how many paricles on random move passing through a little hole ? There is no condition or force acting on eachof them. :biggrin: :biggrin:Thanks guys:biggrin: :biggrin:
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