What is Combination: Definition and 537 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
{\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.
I tried some workings but got me nowhere. I just want to ask whether this question is solvable, i.e the answer can be in numerical value. If yes, then I want to try a bit by myself before asking for hint here.
Thanks
##G## and ##H## are real valued Lipschitz continuous functions. There exists a ##K_1,K_2\geq 0## such that for all ##s,t##,
$$(s-t)^2\leq K_1^2 (G(s)-G(t))^2$$
and
$$(s-t)^2\leq K_2^2 (H(s)-H(t))^2.$$
Is ##aG(t)+bH(t)## where ##a,b## are real constants also Lipschitz continuous?
I tried showing...
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"...
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...
"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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
$\tiny{311.1.3.12}$
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...
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...
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...
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...
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...
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...
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...
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...
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**
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.
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...
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...
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...
Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$
$$a_1\left[
\begin{array}{r}
1\\-2\\0 \end{array}\right],
a_2\left[
\begin{array}{r}
0\\1\\2
\end{array}\right],
a_3\left[
\begin{array}{r}
5\\-6\\8
\end{array}\right],
b=\left[
\begin{array}{r}...
Let's say we have n vectors in ℝ3. And say we have defined a subspace inside ℝ3 in the form of a sphere with radius r, and the center of the spheare is at P, where P is a vector in ℝ3.
What methods exists to find any linear combination of the n vectors, so that the sum of all of them, lies...
Moved from technical forum, so no template is shown
Summary: I have the expression sin(2x) + sin(2[x + π/3]) and I have to write this in terms of a single function (a single harmonic, rather saying). But I don't know how to do this, and... it seems a little bit weird for me, because I'm merging...
Dear all,
I am trying to solve a question, and I think that something is missing.
It is given that the vectors u and v are solutions to the non-homogeneous system of equations Ax=b.
If the vector ku-3v is a solution to the same system, then:
a) k = 4
b) k = 3
c) k = 0
The correct solution...
I can't prove this proposition. I have however managed to prove that the linear combinations of the independent normal rv's are also normal by looking at it's mgf
$$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...
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...