What is Combination: Definition and 534 Discussions

In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.
More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient







(


n
k


)



=



n
(
n

1
)

(
n

k
+
1
)


k
(
k

1
)

1



,


{\displaystyle {\binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}},}
which can be written using factorials as







n
!


k
!
(
n

k
)
!






{\displaystyle \textstyle {\frac {n!}{k!(n-k)!}}}
whenever



k

n


{\displaystyle k\leq n}
, and which is zero when



k
>
n


{\displaystyle k>n}
. The set of all k-combinations of a set S is often denoted by







(


S
k


)






{\displaystyle \textstyle {\binom {S}{k}}}
.
Combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection, k-multiset, or k-combination with repetition are often used. If, in the above example, it were possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears.
Although the set of three fruits was small enough to write a complete list of combinations, this becomes impractical as the size of the set increases. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.

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  1. DaveC426913

    B Simple Combo/Permute calculation

    I'm playing a Steam game called Shapez wherein the goal is to produce and deliver given shapes to the Hub by conveyor belt. In the screencap you can see resources of discs and squares and well as green and red, which are to be extracted, chopped up and recombined to form the "product"...
  2. raminee

    A Combination theory challenge

    Hello All, Not sure if this belongs in general math but lets start here and see where it takes us. In mathematics, a combination is a way of selecting items from a collection where the order of selection does not matter. As an example , say we have digits 1 to 10. And we want to select 3...
  3. hdsfhsdf

    A "The Operation Combination Problem"

    "The Operation Combination Problem" In the "Operation Combination Problem," we explore the space of possible solutions, much like we explore the different possible topologies for the universe. The problem consists of finding all possible combinations of mathematical operations that, when...
  4. milkism

    Method of Images, combination of an infinite plane and a hemisphere

    Problem: I have done part a) in spherical polar coordinates. For part b) I thought it would be just: $$\sigma = -\epsilon_0 \frac{\partial V}{\partial r}$$ But I got confused by "You may want to use different coordinate systems .." So I assume partial derivative w.r.t to r is the spherical...
  5. bo reddude

    Exploring the Math of Glucose Polymers: Understanding 6^n

    Let's say you have n glucose monomers. (C6H12O6) n You want to find out how many possible polymers exist in combining those n number of glucose molecules randomly. So glucose_1 has 6 OHs that can combine with glucose_2 which also has 6 OHs. Starting with glucose_1's first carbon C1, at that...
  6. L

    Combination of 2 formulas for a graph

    i need to combine the two to make a new 3rd one I think the t element is the X axis scale ? not quite sure how to do this, suggestions please ?
  7. Salmone

    I Linear combination of states with Pauli's principle

    If I have two identical particles of ##1/2## spin, for Pauli's exclusion principle all physical states must be antysimmetrical under the exchange of the two particles, so ##\hat{\Pi}|\alpha\rangle=-|\alpha\rangle##. Now, let's say for example this state ##\alpha## is an Hamiltonian eigenfunction...
  8. T

    Venturi effect or Coanda effect or a combination of both in this windmill?

    I have recently find this video and just curious about the working principle behind this concentrated windmill. Apparently it seems that the basic principle is Venturi effect but the shape of the entry point is more suitable for Coanda effect with its curved surface at the entry. And it's also...
  9. M

    A Linear combination of data with uncertainty

    Hello! I have 2 measured data points (they are measurements of different observable, not 2 measurement of the same observable), with quite different errors, say ##x_1 = 100 \pm 1## and ##x_2 = 94 \pm 10##. I want to compute the value (and associated uncertainty) of a linear combination of them...
  10. Lapse

    I How to Combine These Two Probabilities?

    I am trying to determine the likelihood of a driver winning a race based on an associated rating as well as the team he drives for. The probability that Driver A beats Driver B = .8504 The probability that Team A beats Team B = .7576 How do I combine these two probabilities, where the outcome...
  11. C

    One set v is a linear combination of u. Prove u is linearly dependent

    Hi Everybody, I am having some difficulties on the prove this problem. I picked a nice example when I was trying to think about the proof. Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear...
  12. Fobi

    I Probability of 10 consecutive tails with 30 coin flips

    Hi, i was doing a programming exercise that asked me to simulate te flip of coins until it finds 10 consecutive tails. The program usually needs to flips like 6000/8000 coins before finding 10 tails consecutively, but suddenly i found 10 tails with only 30 coin flips, i think that what happened...
  13. greg_rack

    Analyzing a Combination of Beer+Lower Arm for Maximum Acceleration

    Hello guys, to analyse the above-described situation I have opted for considering the body "beer+lower arm" as a whole, therefore computing the new position of the COM for the correct FBD and equations of motion. With some uncertainty on the statement of the problem, I have interpreted the...
  14. M

    I Combination differences with spin

    Hello! I am trying to analyze some diatomic molecular spectra (I am using pgopher) between a ##^2\Sigma_{1/2}## and a ##^2\Pi_{1/2}## level. Before diving into tying to assign lines by eyes in pgopher I was thinking to use this Combination Differences method, but I am not sure I can do it in my...
  15. Shreya

    Design a combination of Capacitors

    My attempt was this: The textbook answer was: Is my approach wrong?
  16. DaalChawal

    MHB Permutation and combination

    Question itself and options 1 and 3.
  17. E

    I am having some difficulties solving a combination circuits problem

    I'm having difficulties finding the current for a 9-ohm resistor only, not the 20-ohm My attempt was to find the voltage of a 20ohm resistor which is V = (0.75A)(20-ohm) = 15V. I got the 0.75A by adding the currents of the two resistors (8-ohm and 16-ohm) together that were in parallel and it...
  18. guyvsdcsniper

    Solving a Linear Combination Problem

    I have attached my work to this thread. Could someone help me with this Linear Algebra problem. This is my first week so I do not know many advanced ways to solve these problems. I could not figure out how to get this matrix into rref, so I solved it the following way. Is the way I used...
  19. DaalChawal

    MHB Permutation and combination

    Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red...
  20. M

    Determine the strength of a Bolt-Nut combination

    Hello! From below sketch, I determined the following assumptions: - Bolt/Nut: M12, Steel 10.9 - The nut is fixed How can I calculate the ultimate force before the bolt will damage the nut? I can imagine the flanges of the threads will break. In documentation I have read about tension surface...
  21. C

    Combination problem with 4 groups of values

    Summary:: 10 values are divided into 4 groups and need a combination of these with restrictions placed on group size, ordering and combinations I have a combinations question.. i have 4 group of values A , B, C and D, with A-2 values, B-3 values , C-2 values, D-3 values. 1. From each group...
  22. P

    I Why the linear combination of eigenfunctions is not a solution of the TISE

    The linear combination of the eigenfunctions gives solution to the Schrodinger equation. For a system with time independent Hamiltonian the Schrodinger Equation reduces to the Time independent Schrodinger equation(TISE), so this linear combination should be a solution of the TISE. It is not...
  23. anemone

    MHB Linear combination of sine and cosine function

    Hi MHB! I recently came across a problem and I was thinking most likely I was missing something very obvious because I couldn't make sense of what was being asked, and I so wish to know what exactly that I failed to relate. Question: Find the minimum of $6\sin x+8\cos x+5$. Hence, find the...
  24. P

    Combination of thin lens and concave mirror

    I created the following ray diagram to help me solve the problem: Then I applied the mirror equation 3 separate times. However, the final image distance I got is wrong. I'm wondering if I'm mistaken in taking the last object distance to be negative. However I only have one more try to get this...
  25. V

    I Combination or Permutation Calculation

    Hello Forum: I have numbers 1 through 6 from which i must select 4 items. The twist is that i need to count only those subsets that include the number 2 all of the subsets are 'distinct' --> 2145 is the same as 2415. My quick calculation yields 15 distinct subsets however some of those do...
  26. karush

    MHB 311.1.3.12 linear combination

    $\tiny{311.1.3.12}$ Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$ $ a_1\left[\begin{array}{r} 1\\0\\1 \end{array}\right], a_2\left[\begin{array}{r} -2\\3\\-2 \end{array}\right], a_3\left[\begin{array}{r} -6\\7\\5 \end{array}\right], b=\left[\begin{array}{r} -7\\13\\4...
  27. K

    A linear combination of states that diagonalize the Hamiltonian

    He told me I "need to show that the Hamiltonian matrix elements you get by using those states have nonzero elements only on the diagonal." I understand what and how a diagonal matrix works, but what I don't understand is what those states are. Are they states I put in my "quantum mechanical...
  28. archaic

    Linear combination of random variables

    a) Total weight ##W=W_1+W_2+...+W_{25}##.$$E[W]=E[W_1]+E[W_2]+...+E[W_{25}]=25\times76=1900\,kg$$$$\sigma_W=\sqrt{V(W_1)+V(W_2)+...+V(W_{25})}=\sqrt{25\times(16)^2}=80\,kg$$ b) Since ##W## is a linear combination of normal distributions, the reproductive property tells us that ##W## is also...
  29. Xavier Labouze

    I Cardinality of Unions of Powersets

    Mentor note: In this thread I (Mark44) have edited "cardinal" to "cardinality." In English, we talk about the "cardinality of a set," not the "cardinal of the set." Given A a set of n elements - note |A| its cardinal and P(A) its powerset. Let A1, A2... Ak, be k subsets (not empty) of A. What...
  30. H

    A Spin-orbital combination and the exclusion principle

    If we combine the two spin-orbital states with ##l=1, s=1/2##, we obtain the combined states with ##J=3/2## and ## J=1/2##. Also, we know that the exclusion principle forbids the two electrons with identical spin-orbital states ##|l,s,m_l,m_s>##. If we combine the two combined states with...
  31. agnimusayoti

    Finding the Fundamental Frequency of a Combination: Boas's Method

    This problem came from Problems, Section 3 Chapter 7 in ML Boas, Mathematical Methods in Physical Sciences. Boas suggested to make a computer plot. From my computer plot (I use online graphing calculator) and find that the period of the sum is 2. Instead of using computer, I want to find the...
  32. iya

    This problem is a combination of optics and derivatives

    According to Cartesian second law, we can get this formula. but wo don't know how to do next. because we just know the sinr and sini.
  33. Lauren1234

    My Too Big? Find Out Here!

    This is my solution however I feel like the number is far too big can anyone see what I’ve done wrong
  34. E

    Calculation involving the reactance of a combination

    If just found out about reactances and impedances today and came across this little problem. I have worked it through with a sort of brute force approach (that I'm not totally sure is correct!) but wondered if it could be done slightly more quickly? I denoted the impedance of the top branch...
  35. tworitdash

    A Waveguide Junction TM/TE combination

    In the paper here, it says no matter what the waveguide structure is, if the smaller waveguide has a TM mode and the larger has a TE mode, then the coupling (Inner cross product \int\int_{S_{smaller-waveguide}} (E^{small}_{transverse} \times H^{large}_{transverse}) . \hat z dS, where z is the...
  36. rachelmaddiee

    Chemistry Cation-Anion Combination: Mg2+ & O2- = MgO

    Magnesium cations (Mg2+) Oxide anions (O 2-) A magnesium ion has a 2+ charge and the oxygen atom has a 2- charge. The formula for the ionic compound is MgO. Would this be correct? Here is my work**
  37. C

    The Combination Of Philosophy And Quantum Physics

    Is there any evidence that Panpsychism may hold its own in the quantum world? What quantum physics principals violate the ideology of Panpsychism? This is meant to be fun. I am curious what other people think.
  38. C

    Why Potential Energy cannot be included twice?

    If for example I have two charged particles q_1 , q_2 with distance 'r' between them, then: The potential energy that results from particle q_1 exerting force on particle q_2 is $$ k\frac{q_1 q_2}{r} $$ If I do the same process for particle q_2: The potential energy that results...
  39. MathematicalPhysicist

    Combination of discrete/continuous signals

    I am looking for literature in theoretical engineering that covers a topic of a signal which is both discrete and continuous. For example ##x[n,t) = t/n## where ##t## ranges over ##[0,\infty)\cap \mathbb{R}## and ##n## is discrete, i.e takes values in ##\mathbb{Z}##. I believe that this isn't...
  40. M

    Combination lens equation questions

    Homework Statement: Lens questions. Homework Equations: idk Hello, so I was wanting to use a laser beam and purchase a few lens' online for a project and wanted to get something similar to the image below, where i would end with a horizontal light ray i was wondering if anyone knows the...
  41. karush

    MHB 1.3.11 Determine if b is a linear combination

    Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$ $$a_1\left[ \begin{array}{r} 1\\-2\\0 \end{array}\right], a_2\left[ \begin{array}{r} 0\\1\\2 \end{array}\right], a_3\left[ \begin{array}{r} 5\\-6\\8 \end{array}\right], b=\left[ \begin{array}{r}...
  42. J

    I Finding a linear combination to enter a sphere

    Let's say we have n vectors in ℝ3. And say we have defined a subspace inside ℝ3 in the form of a sphere with radius r, and the center of the spheare is at P, where P is a vector in ℝ3. What methods exists to find any linear combination of the n vectors, so that the sum of all of them, lies...
  43. Benhur

    Combination of sine functions

    Moved from technical forum, so no template is shown Summary: I have the expression sin(2x) + sin(2[x + π/3]) and I have to write this in terms of a single function (a single harmonic, rather saying). But I don't know how to do this, and... it seems a little bit weird for me, because I'm merging...
  44. Y

    MHB Linear Combination - missing data ?

    Dear all, I am trying to solve a question, and I think that something is missing. It is given that the vectors u and v are solutions to the non-homogeneous system of equations Ax=b. If the vector ku-3v is a solution to the same system, then: a) k = 4 b) k = 3 c) k = 0 The correct solution...
  45. V

    B Linear combination of functions -- meaning?

    h(x) = cf(x) + kg(x) is the linear combination of functions. What makes it linear?
  46. fisher garry

    I Bivariate normal distribution from normal linear combination

    I can't prove this proposition. I have however managed to prove that the linear combinations of the independent normal rv's are also normal by looking at it's mgf $$E(e^{X_1+X_2+...+X_n})=E(e^{X_1})E(e^{X_2})...E(e^{X_n})$$ The mgf of a normal distribution is $$e^{\mu t}e^{\frac{t^2...
  47. T

    MHB Combination of Linear Transformations

    Hello, I'm trying to get my head around linear transformations, and there are a few things I'm not grasping too well. I'm trying to understand combinations of linear transformations, but I can't find a lot of clear information on them. As far as I can tell, any two linear transformations of the...
  48. L

    SI Units: Is the combination of SI units still an SI unit?

    Homework Statement Is N/m^2 a si unit I know Newton is and meters but togeather are there or is m^2 Homework Equations The Attempt at a Solution N/m^2 m^2
  49. Eclair_de_XII

    Finding the distribution of a linear combination of r.v.'s.

    Homework Statement "Let ##X,Y## be independent r.v.'s (EDITED) normally distributed with ##\mu=0,\sigma^2=1##. Find the distribution of ##W=2X-Y##. Homework Equations "If ##X,Y## are independent, then if ##Z=X+Y##, ##f_{Z}=\int_{\mathbb{R}} f_X(x)f_Y(z-x)\, dx##. The Attempt at a Solution...
  50. A

    Combination Circuit Help (pic)

    Could somebody help me out with this combination circuit? The answer that I get is 62K but that is not one of the answers. At first, I thought the 10kohm and 20kohm wherein series and the other side as well. Then I thought they are all in parallel because of the different current flow they all...
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