# Combinations calculator

The calculator finds the number of combinations of the k-th class from n elements without 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 once (without repeated).## Calculation:

$C_{k}(n)=(kn )=k!(n−k)!n! n=10k=4C_{4}(10)=(410 )=4!(10−4)!10! =4⋅3⋅2⋅110⋅9⋅8⋅7 =210$

### The number of combinations: 210

# A bit of theory - the foundation of combinatorics

## 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?

## Foundation of combinatorics in word problems

- Probability 3080

There are eight styles of graduation topics in the Slovak language. The Minister of Education draws 4 of them. What is the probability that he will choose at least one of the pairs? - Probability 80560

I have 3 sources, and their failure probability is 0.1. Calculate the probability that: a) none will have a malfunction b) 1 will have a breakdown c) at least 1 will have a fault d) they will all have a breakdown - Probability 80860

During the exam, the student takes 3 questions out of 20. He is ready for 14 of them. Find the probability that he draws at least one that he knows. - Peaches

There are 20 peaches in the pocket. Three peaches are rotten. What is the probability that one of the randomly picked two peaches will be just one rotten? - Pass a test

The student has to pass a test that contains ten questions. For each of them, he chooses one of 5 answers, with just one being correct. The student did not prepare for the test, so he randomly chose the answers. What are the probabilities that the student - Genetic disease

One genetic disease was tested positive in both parents of one family. It has been known that any child in this family has a 25% risk of inheriting the disease. A family has three children. What is the probability of this family having one child who inher - Seeds

The germination of seeds of a certain species of carrot is 96%. What is the probability that at least 25 seeds out of 30 will germinate? - Bernoulli distribution

The production of solar cells produces 2% of defective cells. Assume the cells are independent and that a lot contains 800 cells. Approximate the probability that less than 20 cells are defective. (Answer to the nearest three decimals). - Syrups

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

What is the probability that in the created trio, which consists of 19 boys and 12 girls, they will be: a) the boys themselves b) the girls themselves c) 2 boys and one girl? - Bouquets

In the flower shop, they sell roses, tulips, and daffodils. How many different bouquets of 5 flowers can we make? - Six questions test

There are six questions in the test. There are three answers to each - only one is correct. To take the exam, students must answer at least four questions correctly. Alan didn't learn, so he circled the answers only by guessing. What is the probability th - Component fail

There is a 90 percent chance that a particular type of component will perform adequately under high-temperature conditions. If the device involved has four such components, determine the probability that the device is inoperable because precisely one of t - Covid-19

Data showed that 22% of people in a small town were infected with the COVID-19 virus. A random sample of six residents from this town was selected. Find the probability that exactly two of these residents were infected.

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