# 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

- Probability - tickets

What is the probability when you have 25 tickets in 5000 that you do not win the first (one) prize? - Dice

How many times must you throw the dice, and the probability of throwing at least one päťky was greater than 50%? - N-gon

How many diagonals have convex 30-gon? - Seating

How many ways can 7 people sit on 5 numbered chairs (e. g., seat reservation on the train)?

- School trip

The class has 19 students. How can students be accommodated in the hostel, where available 3× 2-bed, 3× 3-bed and 1× 4-bed rooms? (Each room has its unique number) - Basketball

Peter and Franta threw to the basket. Each had 20 attempts. Peter scored thirteen and twelve Franta. Give them a percentage of success. - Insurance

The house owner is insured against natural disasters and pays 0.05% annually of the value of the house 88 Eur. Calculate the value of the house. Calculate the probability of disaster if you know that 50% of the insurance is to pay damages. - Football league

In the 5th football league is 10 teams. How many ways can be filled first, second, and third place? - 2nd class combinations

From how many elements can you create 2346 combinations of the second class?

- Confectionery

The village markets have 5 kinds of sweets. One weighs 31 grams. How many different ways can a customer buy 1.519 kg sweets? - Tournament

Determine how many ways can be chosen two representatives from 34 students to school tournament. - Examination

The class is 25 students. How many ways can we choose 5 students for examination? - Bits, bytes

Calculate how many different numbers can be encoded in a 16-bit binary word. - Subsets

How many 19 element subsets can be made from the 26 element set?

more math problems »