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
)

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. chwala

    Solve the given problem involving conditional probability

    Phew! took time to figure this out...i guess there may be a way to use combinations or markov process i do not know... anyway, it was pretty straightforward, we have the ##P_r(w) = \dfrac{n-3}{n}## from box ##X## and this will result in ##P_r(w) = \dfrac{4}{n+1}## in box ##Y##. Together i...
  2. Memo

    Partial Permutation, Combination question for buying dogs at the pet store

    TL;DR Summary: A pet store has 5 Chihuahua, 3 Fox and 4 Poodle. A person wants to buy 7 dogs. How many ways for the person to choose and buy with at least 3 Chihuahua and 2 Poodle. There're C(5,3)*C(4,2) ways to buy dogs with at least 3 Chihuahua and 2 Poodle. There're C(7,2) ways to buy dogs...
  3. N

    Probability using combinations or alternatives

    I have attempted the solution above and I am fairly sure that it is correct. My question is the following: What am I calculating if I multiply 4/52 x 3/51 x 4/50 x 3/49 = 144/ 6,497,400 = 0.002%. I got the correct answer by following the principles of combination probabilities, but intuitively...
  4. C

    Finding possible combinations of capacitors given circuit capacitance

    For this problem, The solution is, Is the only way of finding the possible combinations is by drawing out circuit diagrams? Many thanks!
  5. X

    I don't understand transistors in combinations .... Or maybe basics ....

    Hello everybody ! I do have a problem understanding transistor. I know how these solve and work : I know some basics like the voltage on Rb is Vbb - 0,7V. Or something like this. I also Know that the maximal current in this transistor is "defined" by the resistor in the collector which is...
  6. chwala

    Solve this problem that involves combinations

    My interest is on part b only. I am seeking alternative approach to the problem. This was a tricky question i guess. Find my approach below; ##5C3×4C2 ##{senior cousin included and junior not included}+ ##5C4×4C1##{ senior cousin not included, Junior cousin included}+##5C4×4C2##{both NOT...
  7. johnsmith7565

    Engineering Series-Parallel Combinations of Inductors

    I am not sure where I made the mistake. If someone could point that out that would be much appreciated!
  8. H

    MHB Exploring Your Chocolate Choices: 10 Combinations in 3 Selections

    There are 10 different chocolates and you want to buy three of them. However, you cannot pick a pair of chocolate more than once. How many different choice you can make??
  9. J

    A Number of unequal integers with sum S

    Hello, I've been trying to solve this problem for a while, and I found a technical solution which is too computationally intensive for large numbers, I am trying to solve the problem using Combinatorics instead. Given a set of integers 1, 2, 3, ..., 50 for example, where R=50 is the maximum...
  10. U

    I Limit of limits of linear combinations of indicator functions

    I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...
  11. M

    MHB Game Night : Number of combinations

    Hey! 😊 Sarah, Alex and Mary meet at Tom for a game night. a) The first game is a board game in which six game colors (red, green, yellow, blue, violet, orange) are possible. Each of the players chooses a color in turn. (Each color can only be assigned once.) How many different color...
  12. M

    MHB Counting Color Combinations in 12 Triangles

    There are 12 triangles (picture). We color each side of the triangle in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?
  13. S

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

    A 4×3 matrix which has all elements empty, now I select any two consecutive elements until all elements are selected. I assign an index number (1 to 12) to the matrix element, in one row there are only 1,2,3 elements and 3 & 4 are not consecutive. for example, if I select index 1 & 2 of the...
  14. YouAreAwesome

    How many combinations? (High school math problem)

    Summary:: Year 11 Extension 1 Math problem (Australia) How many combinations can be made from a 4 digit pin code if we can only use two numbers to form our pin code, and we MUST use 2 distinct numbers. E.g. 1112, 4334, 9944, 3232. But NOT 1111, 2113, 0992 etc. We're using the numbers 0-9 and...
  15. V

    B Combinations of 6 taken 4 at a time

    the calculation 6C4 shows 15 but what if all sets are to be distinct? meaning 1,2,3,4 is the same as 4,3,2,1. I made a tree diagram and i get 10... assuming i did that correctly...?
  16. M

    Combinations: exam question distribution

    I think the idea is to first count the number of ways the questions are passed out without restrictions and then subtract the number of ways the two easy questions are together. Call each student ##s_1,s_2,s_3##. We know their total number of questions is 8, so ##s_1+s_2+s_3 = 8 : s_1,s_2,s_3 >...
  17. karush

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

    $ v=\left[\begin{array}{r} -3\\-4\\-5\\4\\-1 \end{array}\right] w=\left[\begin{array}{r} -2\\0 \\1 \\4 \\-1 \end{array}\right] x=\left[\begin{array}{r} 2\\3 \\4 \\-5 \\0 \end{array}\right] y=\left[\begin{array}{r} -2\\1 \\0 \\-2 \\7 \end{array}\right] z=\left[\begin{array}{r} -1\\0 \\2 \\-3...
  18. sahilmm15

    Algebra High school courses on Permutations and combinations

    Can you give me some high school papers or courses on p and c . I have a good source for problems but need a concise and compact course covering concepts. Thanks!
  19. A

    MHB Combinations / Sets of objects

    Hi, I am looking for a solution that generates combinations of objects from a series of objects in a set. For example, {Apple, Pear, Orange} should bring back Apple Pear Orange Apple, Pear Apple, Pear, Orange, Apple, Orange ... Items in the series should not repeat (i.e. Apple, Orange /...
  20. L

    MHB #s of Combinations and Permutations of lines?

    Hello All, See picture below: There exist an infinite plane with infinite number of dots. For sake of argument, let's assume they are 1 inch away from each other. However, below(on your far left) you can see 3 lines already made. The last line is the yellow one. What you see on the left, are...
  21. AN630078

    Probability Questions: Union, Intersection and Combinations

    Question 1: a) T' is the complementary event of T Therefore, T'=1-T In set T = {3,6,9,12} P(T)=4/12 =1/3 P(T')=1-1/3=2/3 b) The addition rule states; P(A ∪ B)=P(A)+P(B)-P(A⋂B) Therefore, P(S ∪ E) = P(S)+P(E)-P(S⋂E) S={1,4,9} P(S)=3/12=1/4 E={2,4,6,8,10,12} P(E)=6/12=1/2 (S⋂E)={4} P(S⋂E)=1/12...
  22. P

    I Closure in the subspace of linear combinations of vectors

    This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the...
  23. R

    License Plate Combinations: Clarifying the Math

    I have a question and searched about at google and found an answer which I don't make sure. If there is 26 letters and 10 digits; my answer is: first letter: 1 way(which is A) second letter: 26 way third letter: 26 way first digit: 1 way(which is 1) second digit 1 way(which is 2) third digit: 10...
  24. M

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

    Proton and neutron are made up of three quarks (uud and udd). Why aren't there particles uuu or ddd?
  25. M

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

    Hi, I'm not sure if this is the correct forum so if I need to post elsewhere please let me know. I'm having trouble with calculating the possible combinations for six digit license plates, numbers 0-9 and letters a-z. I know the overall combinations are 1,947,792 when repetition is allowed...
  26. sergey_le

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  27. C

    Highest value in combinations

    I know nCn/2 = n!/((n/2)!)^2 = 2^(n/2)(1.3.5...n-1) but I am not getting why it's the least one.
  28. Biochemgirl2002

    I How to answer a Combinations question

    Question: When playing Lotto 649, you must pick six numbers from the numbers 1, 2, …, 49. In how many ways can you do this? My attempt n!/r!(n-r)! = 49!/6!(49-6)! (49x48x47x46x...x1)/(6x5x4x3x2x1)(43x42x41x40x...x1) =0.06 . (edit: i redid the question and just made it (49x48x47x46x45x44)/(6!)...
  29. C

    Permutations and Combinations

    the first boy has 6 ways, either he wins 0 or 1 or 2 or 3 or 4 or 5 ways According to the first boy, the second boy also has 0 or 1 or 2 or 3 or 4 or 5 chances According to the first and second boy, the other boy also has 0 or 1 or 2 or 3 or 4 or 5 chances. So, the answer should be 4(either of...
  30. Kaushik

    B Permutations and combinations

    Summary: Is 'Permutation and combination' hard to understand ? I am facing difficulties in understanding this concept. So I would like to ask some questions to enhance my knowledge in this topic. My questions are- Does it require time to grasp this concept? Is it normal to get confused? How...
  31. A

    I Linear combinations of atomic orbitals

    So I've been looking at covalent bonds and come across the approx you can do of the molecular orbital for ##H^+_2## by just summing two 1s orbitals, the method is called the linear combinations of atomic orbitals, and you get what is below which I believe is exact in this case since the 1s...
  32. J

    MHB How to exclude combinations for defined sequences?

    Please can anyone help with the below problem? It’s an interesting problem but please bear with me as i don't have much math background. A factory’s product is sampled once per month every month by its quality inspection team. The factory is allowed up to 2 product failures per ROLLING 12...
  33. Mr Davis 97

    I Affine hull and affine combinations equivalence

    Let ##X = \{x_1 , \dots , x_n\}##. Then ##\text{aff}(X) = \text{intersection of all affine spaces containing X}##. Let ##C(X)## be the set of all affine combinations of elements of ##X##. We want to show that these two sets are equal. First we focus on the ##\text{aff}(X) \subseteq C(X)##...
  34. C

    MHB Find a recursive formula for the number of combinations

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  35. A

    Other Highschool Book for Permutations & Combinations

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  36. C

    MHB Calculating total possible matching combinations remaining

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  37. K

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

    I bought 16 tickets recently and didn't even get 3 digits (out of 6) right. The lotto is from 1 to 58. You choose 6 numbers. The prize is 12 million dollars. How do you compute the probability to get 6 numbers out of 58 numbers? Any tips what combinations to get? I chose many with numbers...
  38. ubergewehr273

    Finding no. of combinations for the situation

    Homework Statement Refer the image Homework Equations Equations for permutations and combinations The Attempt at a Solution Let x be the no. of questions that turned out to be correct. So total score will be 3x-(10-x)=4x-10. The value of this expression must be from the given set and since x...
  39. B

    MHB Puzzle Game - How Many Combinations - How Long To Solve

    For this puzzle game: MillionPuzzleGame.com I'd like to check I worked out the number of combinations correctly. Also, how long would a computer take to solve it?
  40. resurgance2001

    Statistics Permutations and Combinations

    Homework Statement The back row of a cinema has 12 seats, all of which are empty. A group of 8 people including Mary and Francis, sit in this row. Find the number of ways they can sit in these 12 seats if a) There are no restrictions b) Mary and France's do not sit in seats which are next to...
  41. J

    Which of the following combinations would act as buffered solutions

    Homework Statement Which of the following combinations would act as buffered solutions? a. HCl and NaCl b. CH_3COOH and KCH_3COO c. H_2S and NaHS d. H_2S and Na_2S Homework Equations none really... The acid needs to be weak and the base strong though. The Attempt at a Solution HCl is one...
  42. archaic

    B Combinations of n elements in pairs

    Hey! C^2_2 = 1\:\:\:C^2_3 = 3\:\:\:C^2_4 = 6\:\:\:C^2_5 = 10 \\ We\:can\:see\:that\:\:\:C^2_n = C^2_{n-1} + (n-1),\:let's\:try\:n = 5 \\ C^2_5 = C^2_4 + (5 - 1) \\ = C^2_3 + (4 - 1) + (5 - 1) \\ = C^2_2 + (3 - 1) + (4 - 1) + (5 - 1) \\ = 1 + (3 - 1) + (4 - 1) + (5 - 1) \\ C^2_5 = (1 - 1) + (2 -...
  43. Cathr

    I Combinatorics & probability density

    Suppose we have two boxes, each containing three types of balls. On each ball there's written a number: First box: 1, 2, 3 Second box: 4, 5, 6 We don't know how many balls of each type there are, but we know the probability of taking out a specific one, so that we can make a graph showing the...
  44. resurgance2001

    Probability No Member Gets >1 Medal: Math Club

    Homework Statement There are 15 members of a maths club. There are 4 different medals to be randomly given to the members of the club. What is the probability that no member will receive more than one of the medals. Homework Equations Try to find the number of combinations where no member...
  45. W

    MHB Solving Point Distributions for Negotiation Simulation w/Thresholds

    I'm trying to create a negotiation simulation and don't know how to find point allocations for each party that will allow agreement in only about 20% of the cases. Wondering whether anyone here would solve the problem for me. The point allocations I came up with allow agreement only a few...
  46. L

    Exploring Energy Storage Combinations & Optimal Control Schemes

    I'm currently working on combining energy storage systems like flywheels and batteries to balance consumption/regeneration. I've been looking at using an optimal control scheme so that a cost function can be tailored to our wishes. I'm curious about what other people in this field have been...
  47. RoboNerd

    Combination Question on seating

    Homework Statement In how many different ways can you seat 11 men and 8 women in a row if no two women are to sit together? Homework Equations I have the combination and permutation equations The Attempt at a Solution I assume that given the context of this question if I have two, three...
  48. RoboNerd

    Cheese shop combination question

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  49. Z

    Combinations possible when choosing 4 or 5 team members from

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  50. AirRecce

    General formula for a combination of four categories

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