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.
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 -...
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...
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...
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 = []...
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...
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?
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...
// 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...
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...
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?
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...
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?
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...
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.
<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...
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...
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...
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...
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.
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...
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)...
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...
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...
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...
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 =...
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...
hello
I needed to count the number of processes (with the same name). I found that this bash script works,
#!/bin/bash
exit $(ps cax | grep firefox | wc -l);
and the exit value can be caught by this code
#include <stdlib.h> // sytem
#include <iostream> // std::cout, std::endl
int main()
{...
We have an $n \times n$ square grid of dots ($n \ge 2$).
Let $s_n$ denote the number of squares that can be constructed from the grid points.
(a). Show, that $$s_n = \frac{n^4-n^2}{12}.$$
Note, that squares with "diagonal sides" also count.
(b). Evaluate the sum:
\[S = \sum_{k = 2}^{\infty...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...