Combinations without repetition n=4, k=2 result
Find out how many different ways you can choose k items from n items set. With/without repetition, with/without order.Calculation:
Ck(n)=(kn)=k!(n−k)!n! n=4 k=2 C2(4)=(24)=2!(4−2)!4!=2⋅14⋅3=6
The number of combinations: 6
A bit of theory - the foundation of combinatorics
Variations
A variation of the k-th class of n elements is an ordered k-element group formed from a set of n elements. The elements are not repeated and depend on the order of the group's elements (therefore arranged).The number of variations can be easily calculated using the combinatorial rule of product. For example, if we have the set n = 5 numbers 1,2,3,4,5, and we have to make third-class variations, their V3 (5) = 5 * 4 * 3 = 60.
Vk(n)=n(n−1)(n−2)...(n−k+1)=(n−k)!n!
n! we call the factorial of the number n, which is the product of the first n natural numbers. The notation with the factorial is only clearer and equivalent. For calculations, it is fully sufficient to use the procedure resulting from the combinatorial rule of product.
Permutations
The permutation is a synonymous name for a variation of the nth class of n-elements. It is thus any n-element ordered group formed of n-elements. The elements are not repeated and depend on the order of the elements in the group.P(n)=n(n−1)(n−2)...1=n!
A typical example is: We have 4 books, and in how many ways can we arrange them side by side on a shelf?
Variations with repetition
A variation of the k-th class of n elements is an ordered k-element group formed of a set of n elements, wherein the elements can be repeated and depends on their order. A typical example is the formation of numbers from the numbers 2,3,4,5, and finding their number. We calculate their number according to the combinatorial rule of the product:Vk′(n)=n⋅n⋅n⋅n...n=nk
Permutations with repeat
A repeating permutation is an arranged k-element group of n-elements, with some elements repeating in a group. Repeating some (or all in a group) reduces the number of such repeating permutations.Pk1k2k3...km′(n)=k1!k2!k3!...km!n!
A typical example is to find out how many seven-digit numbers formed from the numbers 2,2,2, 6,6,6,6.
Combinations
A combination of a k-th class of n elements is an unordered k-element group formed from a set of n elements. The elements are not repeated, and it does not matter the order of the group's elements. In mathematics, disordered groups are called sets and subsets. Their number is a combination number and is calculated as follows:Ck(n)=(kn)=k!(n−k)!n!
A typical example of combinations is that we have 15 students and we have to choose three. How many will there be?
Combinations with repeat
Here we select k element groups from n elements, regardless of the order, and the elements can be repeated. k is logically greater than n (otherwise, we would get ordinary combinations). Their count is:Ck′(n)=(kn+k−1)=k!(n−1)!(n+k−1)!
Explanation of the formula - the number of combinations with repetition is equal to the number of locations of n − 1 separators on n-1 + k places. A typical example is: we go to the store to buy 6 chocolates. They offer only 3 species. How many options do we have? k = 6, n = 3.
Foundation of combinatorics in word problems
- Simultaneously 80530
The product has a 10% probability of an appearance defect, a 6% probability of a functional deficiency, and a 3% probability of both defects simultaneously. Are the random events A - the product has an appearance defect and B - the product has a functiona - Representative 81580
The chess club has 5 members, including two girls. The circle leader wants to determine by lot which member will represent the circle at the representative tournament. What is the probability that a girl will be drawn? - Probability 1775
The company has so far produced 500,000 cars, of which 5,000 were defective. What is the probability that at most one car out of daily production of 50 cars will be defective? - The probability 2
The probability that an adult possesses a credit card is 0.71. A researcher selects two adults at random. The probability (rounded to three decimal places) that the first adult possesses a credit card and the second adult does not possess a credit card is
- Menu
On the menu are 12 kinds of meals. How many ways can we choose four different meals for the daily menu? - Playmakers 83340
In a basketball game, two pivots, two wings, and one point guard play. The coach has three pivots, four wing players, and two playmakers available on the bench. How many different five players can a coach send to the board during a game? - Probability 3080
There are eight styles of graduation topics in the Slovak language. The Minister of Education draws 4 of them. What is the probability that he will choose at least one of the pairs? - First man
What is the likelihood of a random event where are five men and seven women will first leave the man? - Tokens
The non-transparent bags are red, white, yellow, and blue tokens. We 3times pulled one token and again returned it, writing down all possibilities.
- Olympics metals
How many ways can one win six athletes' medal positions in the Olympics? Metal color matters. - Medals
How many ways can gold, silver, and bronze medals be divided among 21 contestants? - Metals
In the Hockey World Cup, play eight teams, and determine how many ways they can win gold, silver, and bronze medals. - Probability 81685
There are 49 products in the box, of which only 6 are good. We will randomly draw 6 products from them. What is the probability that at least four of the products drawn are good? - Probability 80560
I have 3 sources, and their failure probability is 0.1. Calculate the probability that: a) none will have a malfunction b) 1 will have a breakdown c) at least 1 will have a fault d) they will all have a breakdown
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