Combinatorial calculator
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=10 k=4 C4(10)=(410)=4!(10−4)!10!=4⋅3⋅2⋅110⋅9⋅8⋅7=210
The number of combinations: 210
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
- Trinity
How many different triads can be selected from group 38 students? - Different 68754
We have six balls of different colors. We select two balls at once. How many options? - Determine 79604
In the shoe cabinet, there is one pair each of boots, sandals, tennis shoes, and brown and black ankle boots. Determine how many ways one right shoe and one left shoe can be chosen from among them that do not belong together. - Opportunities 8372
There are 20 students in the class, four of them are being tested by the teacher. How many options are there to choose who the teacher will test?
- 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? - Weekly service
In the class are 20 pupils. How many opportunities has the teacher selected for two pupils who will have a week-class service randomly? - Five brown
Five brown eye contracts, seven green pairs, and four blue pairs. What's the probability Gianna will randomly select a brown or green pair? - Percentages 83188
There are 10 bags displayed in the store, 2 of which have a hidden error. Buyers randomly choose one bag. Express in percentages the probability that they will buy the bag without a mistake. - Disco
At the disco goes 12 boys and 15 girls. In how many ways can we select four dancing couples?
- Sons
The father has six sons and ten identical, indistinguishable balls. How many ways can he give the balls to his sons if everyone gets at least one? - Choices 82334
There are 15 black and 15 white balls in an opaque bag. Elenka took one ball out of the bag three times. what choices of the three balls could she choose? - Three coins
In a game of chance where three coins are tossed, a player wins if two heads and a tail come up. What are the chances of this occurring? - Possibilities 81788
The ring consists of 4 beads. There are 5 different colors of beads in the package. How many possibilities are there to create one ring, and can the colors repeat? - Successively 63644
In an opaque box, identical cubes of different colors: 15 are red, 8 are blue, and 7 are green. We successively drew 10 red, 4 blue, and 3 green dice. What is the probability that we draw a red die from the remaining dice in the next roll?
more math problems »