# What is Combinations: Definition and 414 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
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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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3. ### Probability using combinations or alternatives

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4. ### Finding possible combinations of capacitors given circuit capacitance

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6. ### Solve this problem that involves combinations

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8. ### MHB Exploring Your Chocolate Choices: 10 Combinations in 3 Selections

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9. ### A Number of unequal integers with sum S

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13. ### I Analytical formula for the number of patterns by using combinations?

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14. ### How many combinations? (High school math problem)

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15. ### B Combinations of 6 taken 4 at a time

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16. M

### Combinations: exam question distribution

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17. ### MHB Matrices.......whose null space consists all linear combinations

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18. ### Algebra High school courses on Permutations and combinations

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20. ### MHB #s of Combinations and Permutations of lines?

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21. ### Probability Questions: Union, Intersection and Combinations

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22. ### I Closure in the subspace of linear combinations of vectors

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23. ### License Plate Combinations: Clarifying the Math

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24. ### A Why Proton & Neutron Contain 3 Quarks - Not 2 or 4?

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25. ### MHB Possible combinations for six digit license plates, numbers 0-9 and letters a-z.

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27. ### Highest value in combinations

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29. ### Permutations and Combinations

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32. ### MHB How to exclude combinations for defined sequences?

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33. ### I Affine hull and affine combinations equivalence

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35. ### Other Highschool Book for Permutations & Combinations

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37. ### B Maximize Your Odds: Choosing Winning Lotto Combinations from 1 to 58

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38. ### Finding no. of combinations for the situation

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39. ### MHB Puzzle Game - How Many Combinations - How Long To Solve

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40. ### Statistics Permutations and Combinations

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41. ### Which of the following combinations would act as buffered solutions

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42. ### B Combinations of n elements in pairs

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43. ### I Combinatorics & probability density

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44. ### Probability No Member Gets >1 Medal: Math Club

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46. ### Exploring Energy Storage Combinations & Optimal Control Schemes

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47. ### Combination Question on seating

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48. ### Cheese shop combination question

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49. ### Combinations possible when choosing 4 or 5 team members from

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50. ### General formula for a combination of four categories

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