Counting Definition and 392 Threads

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...

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} =...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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?

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.

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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)...

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...

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

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...

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...

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

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...

$m,n \in N ,and \,\, n\leq 100$ 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 !

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...

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 :)

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!

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...

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...

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"...

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...

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...

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...

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

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...

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...

Are there efficient methods of findings the answer to combination problems like these? The letters A, A, B and C are arranged in any order. How many DISTINCT sequences can we form? If two letters weren't the same, then it would be simple. I can find the answer relatively easily since it's...

Homework Statement 1. 9 women, 6 men are to be seated on a table with 15 seats, how many ways can you assign the seats if no two men are seated next to each other.? 2. How to place 25 unique flags on 10 numbered flagpole if order of the flags on a flagpole is relevant Homework...

Hello, I got problem with this Word problem Determine the number of 7-digit numbers that do not have the consequence 17th For example 4713.1572 allowed 5-digit numbers but 1723,3175,0254 are not allowed. I did a huge progress and get answer 8457 067 but its wrong. I did calculate how many...

This is from Molecular Driving Forces, 2nd Ed. 5.3: Calculating the entropy of mixing. Consider a lattice with N sites and n green particles, and another lattice with M sites and m red particles. These lattices cannot exchange particles. This is state A. (a) What is the total number of...

Probability Question -- using the combinations counting form C(n,k) Homework Statement 4. A bowl contains 3 red chips, 4 green chips, and 5 blue chips. A batch of 3 chips is withdrawn at random. (a) What is the probability that the batch contains only red chips? (b) What is the probability...

This is about exercise 2.20 from Hedman's course. Let me give my solutions to it. (a) $\exists x\exists^{\geqslant 7}y(x\neq y)$ $\exists^{\geqslant 8}x(\varphi(x)\vee\neg\varphi(x))$ (b) $\neg\exists x\exists^{\geqslant 23}y(x\neq y)$ (c) $\exists^{\geqslant 45}x\neg\exists^{\geqslant...

I need to find the statistical error in a counting experiment. Specifically, a decay can proceed via option A or option B and I need to find the branching ratio BR_a=(N_a)/(N_a+N_b). If I were to do this counting experiment multiple times my results for BR_a would follow the binomial...