# Combinatorial calculator

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

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

## Calculation:

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

### The number of combinations: 210

# A bit of theory - the foundation of combinatorics

## 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

- Olympics metals

How many ways can one win six athletes' medal positions in the Olympics? Metal color matters. - Determined 3570

There are 12 points in space, with no three lying on a straight line. How many different planes are determined by these points? - Five-digit

Find all five-digit numbers that can be created from number 12345 so that the numbers are not repeated and then numbers with repeated digits. Give the calculation. - A fair 2

A fair die is thrown two times. What is the probability that the sum of the score is almost 8? - Three shooters

Three shooters shoot, each time, on the same target. The first hit the target with 0.7, the second with 0.8, and the third with 0.9 probability. What is the probability of hitting the target: a) just once b) at least once c) at least twice - Ten dices

When you hit ten dice simultaneously, you get an average of 35. How much do you hit if every time you get six, you're throwing the dice again? - 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? - Cars plates

How many different license plates can a country have since they use 3 letters followed by 3 digits? - Count of triangles

On each side of an ABCD square is 10 internal points. Determine the number of triangles with vertices at these points. - Olympics

How many ways can six athletes be placed on the podium at the Olympics? Depend on the color of the metal. - Hockey game

In the hockey game, they made six goals. The Czech played against Finland. The Czechs won 4:2. In what order to fall goals? How many game sequences were possible during the game? - Hockey match

The hockey match ended with a result of 3:1. How many different storylines may the match have? - Event probability

The probability of event P in 8 independent experiments is 0.33. What is the probability that the event P occurs in one experiment (chance is the same)? - VCP equation

Solve the following equation with variations, combinations, and permutations: 4 V(2,x)-3 C(2,x+ 1) - x P(2) = 0 - 7 heroes

6 heroes galloping on 6 horses behind. How many ways can we sort them behind? - Hockey players

After we cycle, five hockey players sit down. What is the probability that the two best scorers of this crew will sit next to each other?

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