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. 22990atinesh

    Counting One-to-One Functions from n to m with Property f(i)<f(j)

    Homework Statement Let F be the set of one-to-one functions from the set ##{1,2,..,n}## to the set ##{1,2,...,m}## where ##m \geq n \geq 1##. Then how many functions f in F satisfy the property ##f(i)<f(j)## for some ##1 \leq i \leq j \leq n## Homework EquationsThe Attempt at a Solution...
  2. A

    People at the party and counting friends

    Hi 1. Homework Statement On the party came n people. In the beginning all of them have had exactly 3 friends among party members. During party some people made new friends and at the end of the party everyone had exactly 4 friends among party members. Set all numbers n for which the following...
  3. D

    Help Solving Physics Exam Question on Gamma Rays Counting Rate

    Hi, I came across a question in an exam which I couldn't really relate to any topic of physics, that I had studied. It goes like this- A detector is used to count the number of gamma rays emitted by a radioactive source. If the number of counts recorded in exactly 20 seconds is 10000, the...
  4. A

    Calculating the Number of Ways to Make n with k Integers from Given Ranges

    we have to make n with k integers.k integers will have to be choosen from k ranges.Every range has a minimum value and a maximum value.In how many ways we can make n according to the conditions.For example,k=4,n=10 and the ranges are : 1 1 2 2 3 3 4 4 we can make n in only one way.Another...
  5. S

    Band theory electron counting rules for metals/insulators

    I'm a little confused by the description I commonly hear about the electron "counting rule" in band theory. The general statement I find is that a solid with an "odd number of electrons per unit cell is a metal" (because this would imply a partially filled band), while an "even number of...
  6. M

    MHB Counting problem involving numbered cards

    How to solve ii (b) ? Thanks in advance.
  7. Medicol

    Counting edge numbers in bipartite graphs

    Let L be the level number of a bipartite graph G, and so L1 be the first level of n1 vertices, L2 be the second level of n2 vertices, ... Lk be the kth level of nk vertices. Then a bipartite graph G12 is created by a combination of L1 and L2, G23 is of L2 and L3,...,Gij is of Li and Lj. The...
  8. B

    How many elements are in a set of unique rational numbers from 1 to 9?

    Let ##T = \{ \frac{n}{m}\in \mathbb{Q} \vert n, m \in \{ 1, 2, ..., 9 \} \}## No values can repeat (e.g. ##\frac{2}{2},\frac{3}{3},...##) How many elements does the set have. I could just go ahead and count the elements and eliminate the repeats, but I'm wondering if there is a simpler (and...
  9. D

    How to Use Counting Methods and Inclusion-Exclusion for Subsets?

    Homework Statement Let Ω be the universe and A1, A2, A3, ..., An the subsets of Ω. Prove that the number of elements of Ω that belongs to exactly p (p≤n) of the sets A1, A2, A3, ..., An is \sum_{k=0}^{n-p}(-1)^k\binom{p+k}{k}S_{p+k} in which S_{0} = |\Omega| S_{1} =...
  10. Medicol

    Counting squares of NxM lattice

    This is not a quiz but I am thinking how to write down a simple math formula to count the total number of squares present in a lattice of NxM points for my 12 year old nephew ? He'll sure be happy if I could turn this into, say, a common sense for pupils like him. :biggrin: For example, In a...
  11. mishima

    Counting uncertainty in pendulum experiment?

    Hi, I am confused about when the rule for counting uncertainty applies. I know for radioactivity experiments one expresses the uncertainty (error) in the decay count as the square root of the count. So if you counted n decays you would report an average rate of n \pm \sqrt{n} I was...
  12. J

    Sets and Counting: Drug Relief Study Results

    Homework Statement A study was done to determine the efficacy of three different drugs – A, B, and C – in relieving headache pain. Over the period covered by the study, 50 subjects were given the chance to use all three drugs. The following results were obtained: 21 reported relief from drug...
  13. Mogarrr

    Mastering Tricky Counting Problems: Step-by-Step Approach for Guaranteed Success

    I'm trying a few elementary counting problems, and a few are proving very difficult (for me). I have the answers and explanations, which I understand, so that's not the problem. I don't want to memorize answers. The problem is systematically analyzing these problems. My intuition is almost...
  14. I

    Counting the distinct values of a modular mapping

    Hello, first of all, sorry if my question is either trivial or imprecise, I'm from the engineering domain :) I need to know how many different values the following pair can take: \left(a\cdot i + b\cdot j\right) \bmod n_1 \left(c\cdot i + d\cdot j\right) \bmod n_2 as (i,j) spans \mathbb{Z}^2...
  15. P

    Double Counting Correction in LDA+U: AMF vs. FLL

    Hi there, I'm not sure if someone here can help me because the topic is quite specific. In LDA+U Calculation one have to substract the Double Counting from the DFT functional. There are two main possibilities to do this: 1.) Around Mean Field (AMF) 2.) Fully Localized Limit (FLL) I...
  16. 22990atinesh

    Counting total Number of Functions

    Homework Statement Consider Sets A={1,2,3,...,n} and B={1,2,3,...,m} where 1<=n<=m. F:A->B 1) Total no. of strictly increasing function i.e if x<y then f(x)<f(y) 2) Total no. of non decreasing function i.e if x<y then f(x)<=f(y) How can we count total number of functions here 2. The attempt...
  17. Y

    MHB Counting problem - Multiple choice test

    A quiz has 4 questions with 3 choices for each answer. If you guess every answer, in how many different ways can you complete this test?__________ How many students must take this test to guarantee that at least 3 identical answer sheets are submitted?__________ I know how that the answer to...
  18. N

    Counting problem: 5-character ASCII strings containing at least one @

    Homework Statement How many strings of five ASCII characters contain the character @ ("at" sign) at least once? [Note: there are 128 different ASCII characters.] Homework Equations The rule of product and inclusion-exclusion principle are relevant. The Attempt at a Solution The correct...
  19. N

    Counting Dark Fringes: Measurement & Order

    Dark spots are measured on a screen at +xm, +ym and +zm where z>y>x from the central axis. since dark fringes are where (m+0.5)λ, would it be right for me to state that the dark spot z has an order of 3.5?
  20. O

    MHB Counting Cosets: Clarifying Right & Left Cosets

    i am reading a chapter on counting cosets and I am not sure i fully understand the theory behind right and left cosets. can i please be given clear descriptions perhaps with examples.
  21. K

    Counting degrees of freedom for Goldstone bosons

    I mean Goldstone bosons in the title. Sorry I don't know how to edit the title. Goldstone's Theorem says that there is a massless Goldstone mode for each breaking symmetry. For instance symmetry of a theory is broken from SU(N) to SU(N-1), the # of Goldstone bosons is (N^2-1)-((N-1)^2-1)=2N-1...
  22. C

    Probability of counts from the counting rate of a radioactive sample?

    Homework Statement The average counting rate of a radioactive sample is 486 cpm (counts per minute). Find the probability that in any given 10s interval one gets less than 72 counts. Is this the same as the probability of getting less than 72 X 6 = 432 counts in 60s? Homework Equations...
  23. X

    Is my equation for counting primes unique or similar to existing equations?

    Dear fellow learners, Through an extracurricular project I have found a really cool equation to count primes. The equation can evaluate Pi(x)+Pi(√x)/2+Pi(cubedroot(x))/3+...Pi(nthroot(x))/n I have directly proved my equation so I now it will be accurate 100% of the time. Although the...
  24. L

    Solve the SYLLABUS Word Problem with Combinations | Homework Help

    Homework Statement Calculate the numbers of different selections of 5 letters which can be made from the letters of the word SYLLABUS Homework Equations The Attempt at a Solution I thought it would be 8C5 but that's not the answer. Any help?
  25. Y

    MHB Solve 17 Difficult Math Problems: Proving At Least 2 Committees are Identical

    There are 4 people in a class. Among them, they are to solve 17 difficult math problems. They form 17 committees – one committee to deal with each of the 17 problems. Prove that at least 2 of these committees contain exactly the same people. PLEASE EXPLAIN HOW EACH STEP TO REACH TO THE...
  26. L

    Counting Combinations: How to Divide 4 Articles Between 2 People | Homework Help

    Homework Statement In how many ways can 4 articles be divided between 2 people when each person must receive at least one article? Homework Equations The Attempt at a Solution I tried it like this Person 1 could have ( 3,2 or 1 paper(s) ) Person 2 the same so total # ways...
  27. L

    Seating Arrangements in a Railway Compartment

    Homework Statement There are 8 seats in a railway compartment. In how many ways can 8 people be seated if 2 must have their backs to the engine and 1 must face the engine? Homework Equations The Attempt at a Solution So 3 of them must be in those exact positions so I tried...
  28. L

    Counting Question: At Least 12 Hearts in 13-Card Hand

    Homework Statement How many different hands of 13 cards can be obtained containing at least 12 hearts? Homework Equations The Attempt at a Solution I'm confused, well I know there could be a possibility of getting 13 hearts, but I am getting confused trying to work out the other...
  29. C

    Counting Ways to Place 8 Rooks on a Chess Board without Attacks

    Homework Statement How many ways can you place 8 rooks on a chess board so that no 2 rooks attack each other. Assume the rooks are identical. Chess board is 8 by 8. The Attempt at a Solution To place the first rook I would have 64 choices. for the second rook I would have 49 choices...
  30. M

    Gibb's phase rule and counting phases

    Hello, I was wondering, suppose I had a mixture of a few liquids that were immiscible, say 4. Now, when calculating the degrees of freedom, I am confused about the number of phases. I think the number of components is 4, but the number of phases should be 1, liquid, right? Or does each...
  31. D

    Optimal Itinerary Planning for a Business Trip to Six Major Cities

    Homework Statement A businesswoman in Philadelphia is preparing an itinerary for a visit to six major cities. The distance traveled, and hence the cost of the trip, will depend on the order in which she plans the route. a) how many different itineraries ( and trip costs) are possible)...
  32. N

    MHB Math final review: counting problems

    Can somebody please give me a heads up on if I solved these two problems correctly, I appreciate it, thank you! d) In setting up a new department, a corporation executive must select a manager from among 4 applicant, 3 clerks from among 9 applicants, and 2 secretaries from among 7 applicants...
  33. M

    Discover the Number of 8 Letter Words with 3 Vowels and Non-Repeating Consonants

    How many 8 letter words have 3 vowels constants do not repeat. Is it 5^3*26*25*24*23*22 ?
  34. M

    Counting problem (book wrong?)

    Question and my solution is in the paint document. Case 1 - Case 3 represents all possible combinations of string of length 4 with exactly three 9's. The d represents a digit that is not 9. d = 0 or 1 or 2 or 3 or ... or 8: This is 9 options. Notice that none of the cases 1 - 3 have identical...
  35. M

    Permutation/Combination Theorem: Solving Problem 46b & 47a

    Question 1: My question is how do I know when to use the permutation theorem or comination theorem? Theorem in paint document I'm asking because I would have used permutation thm. on 46 but I was wrong. Problem 46 b. and Problem 47 a. seem similar. The two main differences...
  36. R

    How Many Images Does a Zone Plate Produce with an Object in Front?

    How many images does a zone plate produces when an object is placed infront of zone plate?
  37. A

    Cantor set ℵ , inductive proofs by openly counting.

    I have been looking at the idea of 1:1 correspondence as a method of determining set size/cardinality, and have noticed that the principle allows for inductive proofs, which I think are properly constructed, that can come to conclusions which are clearly wrong under traditional set theory if...
  38. Albert1

    MHB The student made a mistake in his counting

    $m,n \in N ,and \,\, n\leq 100$ a student counts : $\dfrac {m}{n}=A.a_1a_2a_3--------a_k167a_{k+1}---$ please prove : the student's answer is not correct , there must have a mistake in his calculation !
  39. M

    Solve the P6 Counting Problem: Unlock 10*365 Password Combinations

    Im trying to find all combinations of P6. Book solution in paint doc. My solution: Please tell me where I am going wrong. P6: Password of 6 characters 1. Each password must contain at least one digit, 2. Each character of password can be a digit or uppercase letter. Let P61 be defined...
  40. M

    Counting Elements in Sets: Steps Included

    How many elements does each of these sets have where a and b are distinct elements? (with steps please) a) P({a,b{a,b}})b)P({∅,a,{a},{{a}}}) c)P(P(∅)) *i have tried to solve them but i am a little bit confused... Thanks in advance :)
  41. T

    MHB Calculate Probability of Y or More People in N Train Cars

    Quick question for you all... If I had X number of train cars and I wanted to know the probability of having Y or more people in a car when I have N total of people. How would I go about solving this? Thanks!
  42. A

    Counting Combos: 5 T-Shirts & 3 Jeans: 15 Days

    A student has five different t-shirts and three pairs of jeans. How many days can the student dress without repeating the combination of jeans & t-shirt? How many days can the student dress without repeating the combination of jeans & t-shirt and without wearing the same t-shirt on two...
  43. MI5

    MHB Counting proof of the addition rule

    Let $ \left\{A_1, A_2, \cdots , A_n\right\}$ be a system of subsets of a finite set $A$ such that these subsets are pairwise disjoint and their union $A = \cup_{i=1}^{n}A_{i}$. Then $ |A| = \sum_{i=1}^{n}|A_i|$. (1) Proof: According to the hypothesis, each $a \in A$ belongs to exactly one of...
  44. trash

    [Pascal] Counting letters in a row without arrays

    I'm working with this excercise: make a program that gives the number of consecutive letters of a word and the most repeated letter in a row -example: if i enter aaafdseergftth and i press return the program should return a = 3, e=2, t=2 and a=3-. I've come up with a couple of "solutions"...
  45. P

    Not Linear counting, 3D counting billions of billionths

    Many of the numbers that define us and our world are beyond the scope of linear counting. If you count a billion seconds, it is more than 32 years, but if you can visualize them as a cube, it has sides of 16 minutes and 40 seconds. A billion millimetre cubes in a line span 1000km, but if you put...
  46. reenmachine

    Set Theory - Counting - Binomial Coefficient - Factorials

    Homework Statement A department consists of 5 men and 7 women.From this department you select a committee with 3 men and 2 women.In how many ways can you do this? Homework Equations Since the "overall set" (the entire department) is composed of both men and women and each has a specific...
  47. reenmachine

    Factorials and lists/subsets counting

    1.1 Homework Statement Using only pencil and paper , find the value of ##\frac{120!}{118!}## 2.1 Relevant equations ##\frac{120 \cdot 119 \cdot 118!}{118!} = 120 \cdot 119 = 14280## 1.2 Homework Statement Compute how many 9-digit numbers can be made from the digits...
  48. anemone

    MHB Divisibility and Digit Counting: Solving the Five-Digit Number Challenge

    How many five digit numbers are divisible by 3 and contain the digit 6?
  49. P

    Verifying 226-Ra Half-Life Calculation

    Homework Statement I think I'm very right om this assignment, but I would like to completely sure - so I'm thankful if someone educated in nuclear physics can check that this is correct. 226-Ra has a half-life of 1600 years. One source of radiation contains 1.0 mg of this Radium nuclide...
  50. E

    Counting microstates in Boltzmanns principle

    Homework Statement Explain why the number of microstates W in Boltzmanns principle, is W = ƩNi! / ∏Ni! when i ideal gasses are mixed at constant volume and temperature. Ni is the number of particles of component i. Homework Equations S=klnW , where W is the number of microstates...
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