Combinations without repetition n=12, k=5 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=12 k=5 C5(12)=(512)=5!(12−5)!12!=5⋅4⋅3⋅2⋅112⋅11⋅10⋅9⋅8=792
The number of combinations: 792
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
- Calculation of CN
Calculate: (486 choose 159) - (486 choose 327) - Cards
The player gets eight cards of 32. What is the probability that it gets a) all four aces b) at least one ace - Student examination
How many ways can a teacher select a group of 6 students to sit in the front row if the class has 13 students? - Four-digit 79614
Determine the number of all four-digit natural numbers in decimal notation in which the digit 0 is not present, and each of the remaining nine numbers occurs at most once. - Contestants 67104
The contestants have to create an ice cream sundae containing three different types of ice cream. They can use cocoa, yogurt, vanilla, hazelnut, punch, lemon and blueberry ice cream. How many different ice cream sundaes can the contestants create? - Probabilities
If probabilities of A, B, and A ∩ B are P (A) = 0.62, P (B) = 0.78, and P (A ∩ B) = 0.26, calculate the following probability (of the union. intersect and opposite and its combinations): - Probability 80856
The probability of occurrence of a certain phenomenon is the same in all trials and is equal to 0.7. Attempts are repeated until this phenomenon occurs. What is the probability that we will have to make a fifth trial? - First class
The shipment contains 40 items. 36 are first-grade, and four are defective. How many ways can we select five items so that it is no more than one defective? - Intersection of the lines
How many points do nine lines intersect in a plane, of which four are parallel, and of the other five, no two are parallel (and if we assume that only two lines pass through each intersection)? - Menu choice
In a Jollibee, you have a menu choice of C1, C2, and C3. For dessert, you have a choice of ice cream and mango peach. How many different options do you have? - Dice
We threw ten times playing dice. What is the probability that the six will fall exactly four times? - Probability 69714
The factory produces 35% of the tiles on line A, which produces scrap with a probability of 0.02, and 65% on line B, where the probability of scrap is 0.03. What is the probability that the selected tile will be defective? - Six on dice
When throwing two dice, what is the probability they will fall at least one six? - Four-digit 65124
Please find out how many different four-digit numbers we can create from the digits 3 and 8 so that the two digits three and two digits eight are used in each four-digit number created. - Probability 71784
What is the probability that if you roll the die twice, the sum of 12 will fall?
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