n choose k calculatorFind out how many different ways you can choose k items from n items set without repetition and without order. This number is also called combination number or n choose k or binomial coefficient or simply combinations. See also general combinatorial calculator.
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
Number of combinations: 210
A bit of theory - foundation of combinatorics
VariationsA 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, equivalent. For calculations, it is fully sufficient to use the procedure resulting from the combinatorial rule of product.
PermutationsThe 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 repetitionA 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 repeatA 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.
CombinationsA 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 repeatHere 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
- Probability bullets
There are six red, five green, eight blue, 11 yellow balls in the bag. What is the probability that we will pull out the green bullet?
- Permutations without repetition
From how many elements, we can create 720 permutations without repetition?
- Genetic disease
One genetic disease was tested positive in both parents of one family. It has been known that any child in this family has a 25% risk of inheriting the disease. A family has 3 children. What is the probability of this family having one child who inherited
- Right key
The hostel has 4 rooms. The keys to each room are not numbered. Each of the four guests took one key. What is the probability that everyone took the right key?
- Three shooters
Three shooters shoot, each one time, on the same target. The first hit the target with a probability of 0.7; second with a probability of 0.8 and a third with a probability of 0.9. What is the probability to hit the target: a) just once b) at least once c
- Classical 69634
Peter, Jano, Alice, and Rebecca went to a classical music concert. How many different ways can they sit in the four free seats if Rebecca wants to sit with John?
- 2nd class variations
From how many elements you can create 2450 variations of the second class?
- Three robots
In a workshop, three robots, Q, R and S, are employed to make chairs Robot Q makes 25% of the chairs Robot R makes 45% of the chairs The remaining chairs are made by Robot S Evidence has shown that 2 percent of the chairs made by robot Q are d
- Component fail
There is a 90 percent chance that a particular type of component will perform adequately under high temperature conditions. If the device involved has four such components, determine the probability that the device is inoperable because exactly one of the
- Three-digit integers
How many three-digit natural numbers exist that do not contain zero and are divisible by five?
- An aircraft
An aircraft manufacturing company has submitted bids on two separate airline contracts, A and B. The company feels that it has a 70% chance of winning contract A and a 25% chance of winning contract B. Furthermore, it believes that winning contract A is i
How many five-digit numbers can be written from numbers 0.3,4, 5, 7 that is divided by 10, and if digits can be repeated?
- Probability 3219
In recent years, it has rained 12 days in March. What is the probability that it rained on March 18?
- The test
The test contains four questions, and there are five different answers to each of them, of which only one is correct, the others are incorrect. What is the probability that a student who does not know the answer to any question will guess the right answer
- Three-digit numbers
Use the numbers 4,5,8,9 to write all three-digit numbers without repetition. How many such numbers are there?
- Squares above sides
In a right triangle, the areas of the squares above its sides are 169; 25 and 144. The length of its longer leg is:
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