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







(


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. Xavier Labouze

    I Cardinality of Unions of Powersets

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  3. C

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  4. Suyash Singh

    B Permutations and combinations

    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...
  5. S

    B Linear combinations

    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...
  6. H

    8 balls how to arrange for adjoining?

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  7. K

    How many ways can 12 balls be arranged into 4 different rows

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  8. M

    Need help with counting problems

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  9. W

    How many ways can you arrange 52 things into 4 groups BUT th

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  10. F

    Ways to put letters in postboxes

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  11. P

    Algorithm for creating unique groups of elements

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  12. K

    Proving combination is a natural number by induction

    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
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