# Combinations without repetition n=65, k=8 result

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

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

## Calculation:

$C_{k}(n)=(kn )=k!(n−k)!n! n=65k=8C_{8}(65)=(865 )=8!(65−8)!65! =8⋅7⋅6⋅5⋅4⋅3⋅2⋅165⋅64⋅63⋅62⋅61⋅60⋅59⋅58 =5047381560$

### The number of combinations: 5047381560

5047381560

# 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

- Trinity

How many different triads can be selected from group 38 students? - Opportunities 8372

There are 20 students in the class, four of them are being tested by the teacher. How many options are there to choose who the teacher will test? - 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 - Disco

At the disco goes 12 boys and 15 girls. In how many ways can we select four dancing couples? - Distribution 2645

The worker operates 600 spindles on which the yarn is wound. The probability of tearing the yarn on each spindle 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) What is t - Orchard

10 trees in 5 lines grow in the orchard. How many trees are in the orchard? - Possibilities 81788

The ring consists of 4 beads. There are 5 different colors of beads in the package. How many possibilities are there to create one ring, and can the colors repeat? - Cards

How many ways can you give away 32 playing cards to 7 player? - Numbers 72404

How many numbers are less than 200, the digits sum of which is 6? - Fourland 3542

In Fourland, they only have four letters F, O, U, and R, and every word has exactly four letters. No letter may be repeated in any word. Write all the words that can be written with them. - travel agency

A small travel agency offers five different tours on honeymoon. What is the probability that the bride and groom choose the same tour (they choose independently)? - Repetition 80362

How many six-digit numbers without repetition can be formed from the digits 1, 2, 3, 4, 5, and 6, if the numbers are, to begin with: a) the digit 4; b) digits 4 or 5? - Positions 26151

How many positions are there to store three books on the shelf? - Ruben

Ruben owns a restaurant. He likes to keep track of everything customers are buying. His top 3 sellers are sandwiches, salads, and pizza. He knows that 1/3 of his customers buy a sandwich, 1/2 buy a salad, and 1/4 buy a pizza. What fraction of customers bu - Gold, silver, bronze

How many ways can we divide gold, silver, and bronze medals if six people compete?

more math problems »