combination Definition and Topics - 12 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
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{\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
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{\displaystyle \textstyle {\binom {S}{k}}}
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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.
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 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...
The total number of different combinations of one or more letters which can be made from the letters of the word MISSISSIPPI is?
First i dont understand what the question means
and second my answer is completely different from that in my book
My working-
since there are 11 letter 4 I 4 S 2 P...
Hi, I read that linear combinations of a state, Psi, can be as:
\begin{equation}
\Psi = \alpha \psi + \beta \psi
\end{equation}
where ##\alpha## and ##\beta## are arbitrary constants.
Can however this be a valid linear combination?
\begin{equation}
\Psi = \alpha \psi \times \beta \psi...
Homework Statement
Homework Equations
The Attempt at a Solution
the answer for no adjoining
_W_W_W_W_W_
for 3 red balls, there are 6 positions
so ## 6C_3 = 20##
i'm curious, on other way to find arrangement?
for adjoining = all arrangement - adjoining
all arrangement = 3 red can get to...
Homework Statement
In how many ways can 12 balls be arranged into 4 different rows with each row having
at least one ball
(a) if the balls are identical?
(b) if there are 6 identical red balls and 6 identical blue balls?
Homework Equations
The Attempt at a Solution
a)
Put 4 balls in each...
Homework Statement
Counting problems are a very tough subject to me, so if someone could give me tips, examples explaining what's really happening, that would be great.
Homework Equations
I know what permutations, variations, combinations, ... are. The problems involving only one of those...
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combination
counting problem
permutation
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Homework Statement
There are 7 different postbox, and 10 identical letters. How many ways can the letters put into the boxes so that there is at least one letter in a postbox?
Homework Equations
nCr=n!/(n-r)!r!
If M,N,O..... things can be done in m,n,o..... ways then ways of doing them...
Homework Statement
so for a side task I'm supposed to assign people to groups for an icebreaker in python, can anyone give me links to theories that I could read up on or give me suggestion
X number of people at my company signed up for a dinner roulette as a way to meet new people. Everyone...
Hi,
I've seen on on several sites that you can prove that nCr, where r<=n, is a natural number. I'm not sure how to do this by induction.
So I need help on this proof. How do I write this as a mathematical statement at the start of the induction proof?
Thank you