# Combinations with repetition

Find out how many different ways you can choose*k*items from

*n*items set. With/without repetition, with/without order.

## Calculation:

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

### Number of combinations with repetition: 286

# A bit of theory - 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, 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

- Trinity

How many different triads can be selected from group 38 students? - School parliament

There are 18 boys and 14 girls in the class. In how many ways can 3 representatives be elected to the school parliament, if these are to be: a) the boys themselves b) one boy and two girls - Disco

At the disco goes 12 boys and 15 girls. In how many ways can we select four dancing couples? - Cards

How many ways can you give away 32 playing cards to 7 player? - Orchard

10 trees in 5 lines grows in the orchard. How many trees are in the orchard? - Distribution 2645

The worker operates 600 spindles on which the yarn is wound. The probability of tearing of the yarn on each of the spindles at time t is 0.005. a) Determine the probability distribution of the number of torn spindles at time t and the mean and variance. b - Numbers 72404

How many numbers are less than 200, the digits sum of which is 6? - travel agency

Small travel agency offers 5 different tours at honeymoon. What is the probability that the bride and groom choose the same tour (they choose independently)? - Positions 26151

How many positions are there to store three books on the shelf? - Gold, silver, bronze

How many ways can we divide gold, silver, bronze medals if 6 people are competing? - Football league

In the football league is 16 teams. How many different sequences of results may occur at the end of the competition? - Bits, bytes

Calculate how many different numbers can be encoded in 16-bit binary word? - Ball bearings

One bearing is selected from the shipment of ball bearings. It is known from previous deliveries that the inner bearing radius can be considered as a normal distribution of N (µ = 0.400, σ2 = 25.10^−6). Calculate the probability that the selected radius w - Four-member team

There are 14 girls and 11 boys in the class. How many ways can a four-member team be chosen so that there are exactly two boys in it? - Combinations 6

6 purses 9 flaps 12 straps Every combination must include 1 purse, 1 flap, and 1 strap. How many are possible combinations? - Party

At the party everyone clink with everyone. Together, they clink 406 times. How many people were at the party?

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