# Combinations with repetition

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:

$C_{k}(n)=(kn+k−1 )n=10k=4C_{4}(10)=C_{4}(10+4−1)=C_{4}(13)=(413 )=4!(13−4)!13! =4⋅3⋅2⋅113⋅12⋅11⋅10 =715$

### The number of combinations with repetition: 715

# A bit of theory - the foundation of combinatorics

## 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

- Sons

The father has six sons and ten identical, indistinguishable balls. How many ways can he give the balls to his sons if everyone gets at least one? - Two groups

The group of 10 girls should be divided into two groups with at least four girls in each group. How many ways can this be done? - Three workplaces

How many ways can we divide nine workers into three workplaces if they need four workers in the first workplace, 3 in the second workplace, and 2 in the third? - Cinema

How many ways can 11 free tickets to the premiere of "Jáchyme throw it in the machine" be divided between 6 pensioners? - Three reds

What is the probability that when choosing 3 carats from seven carats, all three reds will be red? - Kenneth 2

Kenneth has 100 pennies, 20 nickels, 10 dimes, and 4 quarters. How many ways can he choose coins that total 25 cents? - Boys and girls

There are 11 boys and 18 girls in the classroom. Three pupils will answer. What is the probability that two boys will be among them? - Candy

How many ways can divide 10 identical candies to 5 children? - Six attractions

How many opportunities do you have if you want to complete ten rides on the fair, but there are only six attractions? - Cards

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

We want to plant five fruit trees in the garden, of which three are apple trees and two pears. How many different ways can we organize them? - The gems

The jeweler selects four gems for the ring. It has rubies, emeralds, and sapphires. How many options does he have? - Honored students

Of the 25 students in the class, ten are honored. How many ways can we choose five students from them if there are to be exactly two honors between them? - Sapphires 45461

The jeweler selects three gems in the ring. It has rubies, emeralds, and sapphires. How many choices does it have? - How many 2

How many three-lettered words can be formed from letters A B C D E G H if repeats are: a) not allowed b) allowed? - Syrups

In the shop, they sell three types of syrups - apple, raspberry, and orange. How many ways can you buy four bottles of syrup?

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