# 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

- Calculation of CN

Calculate: (486 choose 159) - (486 choose 327) - Soccer teams

Have to organize soccer teams. There are three age groups. How many different ways can you organize ten teams for each age group? Is this a permutation or combination? - Chords

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

The player gets eight cards of 32. What is the probability that it gets a) all four aces b) at least one ace - 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? - School parliament

There are 18 boys and 14 girls in the class. In how many ways can three representatives be elected to the school parliament if these are to be: a) the boys themselves b) one boy and two girls - Contestants 67104

The contestants have to create an ice cream sundae containing three different types of ice cream. They can use cocoa, yogurt, vanilla, hazelnut, punch, lemon and blueberry ice cream. How many different ice cream sundaes can the contestants create? - Probability 80856

The probability of occurrence of a certain phenomenon is the same in all trials and is equal to 0.7. Attempts are repeated until this phenomenon occurs. What is the probability that we will have to make a fifth trial? - Intersection of the lines

How many points do nine lines intersect in a plane, of which four are parallel, and of the other five, no two are parallel (and if we assume that only two lines pass through each intersection)? - Hockey

The hockey match ended 7:4. How many different matches could be? - First class

The shipment contains 40 items. 36 are first-grade, and four are defective. How many ways can we select five items so that it is no more than one defective? - Combinatorics

The city has 7 fountains. Works only 6. How many options are there that can squirt? - Combinations

Evaluate following expression involving combinations and permutations: C(6,3) + 3 P(6,3) - Dice

We threw ten times playing dice. What is the probability that the six will fall exactly four times? - Points in plane

The plane is given 12 points, 5 of which are located on a straight line. How many different lines could be drawn from these points?

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