# Combinations with repetition n=11, k=3 result

The calculator finds the number of combinations of the k-th class from n elements with repetition. A combination with repetition of k objects from n is a way of selecting k objects from a list of n. The order of selection does not matter and each object can be selected more than once (repeated).## Calculation:

$C_{k}(n)=(kn+k−1 )n=11k=3C_{3}(11)=C_{3}(11+3−1)=C_{3}(13)=(313 )=3!(13−3)!13! =3⋅2⋅113⋅12⋅11 =286$

### The number of combinations with repetition: 286

# 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 V

_{3}(5) = 5 * 4 * 3 = 60.

$V_{k}(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:$V_{k}(n)=n⋅n⋅n⋅n...n=n_{k}$

## 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.$P_{k_{1}k_{2}k_{3}...k_{m}}(n)=k_{1}!k_{2}!k_{3}!...k_{m}!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:$C_{k}(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:$C_{k}(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

- Shooters

In the army, regiments are six shooters. The first shooter target hit with a probability of 49%, next with 75%, 41%, 20%, 34%, 63%. Calculate the probability of target hit when shooting all at once. - Fish tank

A fish tank at a pet store has 27 zebrafish. In how many different ways can George choose 2 zebra fish to buy? - 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? - Calculation of CN

Calculate: (486 choose 159) - (486 choose 327) - Raspberries 66824

Klára wants to make a fruit cocktail from three types of fruit. It has pineapple, pears, bananas, raspberries, and cherries. How many different cocktails can he create? - 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 - Trinity

How many different triads can be selected from group 38 students? - Designated 66594

Marenka is required to read three books out of five designated books. How many ways can three books choose to be read? - How many 32

How many ways can a teacher select a group of 6 students to sit in the front row if the class has 13 students? - Menu

On the menu are 12 kinds of meals. How many ways can we choose four different meals for the daily menu? - 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? - Five-a-side 69434

Five children took part in the five-a-side tournament: Anka, Betka, Celeste, Dano, and Erik. Everyone played with everyone. How many games have been played? - In PE

In PE, students play a game where they do different exercises depending on the color of marble that Coach Forbes draws. Coach Forbes has a jar with 6 red marbles, 12 blue marbles, 16 purple marbles, two green marbles, and four yellow marbles. What is the - Chords

How many 4-tones chords (chord = at the same time sounding different tones) is possible to play within 7 tones? - How many 31

How many ways can a teacher select a group of 3 students to sit in the front row if the class has 13 students?

more math problems »