# Combinations Definition and 19 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. ### A Number of unequal integers with sum S

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

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3. ### I Combinatorics & probability density

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5. ### Combination Question on seating

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6. ### Cheese shop combination question

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

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8. S

### Probability (Permutation Combination)

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9. ### B Arranging blocks so that they fit together

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10. ### 8 balls how to arrange for adjoining?

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11. M

### Combinatorics: tennis game with 8 people

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12. ### I How many combinations of unique arrangements are there?

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13. ### Combinatoric problem

We have a square grid. In how many ways we get to the point [20,30], if we can not get points [5,5], [15,10] [15,10] [17,23]? The starting point is [0,0], we can only move up and to the right. Thanks My solutions:  \binom{50}{20} - \binom{10}{5}\binom{40}{15} - \binom{25}{10}\binom{25}{10}...
14. ### Algebra Books for learning multinomial theorem

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15. ### Number of ways in a 3D lattice

Hi! If I have points A and B in a lattice in the plane, and the closest path between them is n + m steps (for example 4 steps upwards and 5 steps to the right), there are C(9,(5-4)) = 9 combinations of paths between them. I have to choose the 4 ways upwards (or the 5 ways to the right) of the 9...
16. ### Combinations of three players

Hi, I am trying to find the solution (formula) for doing the following: For any number of players (divisible by three), I need to calculate a number of rounds (in my case 8) for three players to play each other without any repetition. So the first round of etc. 21 players would be: 1,2,3 4,5,6...
17. ### Probability problem

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18. ### Prime factors of binomials

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19. ### Permutations and combinations - is square a rectangle?

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