# Variations without repetition

The calculator calculates the number of variations of the k-th class from n elements. Variation is a way of selecting k items from a collection of n items (k ≤ n), such that (like permutations) the order of selection does matter. The repetition of items is not allowed.## Calculation:

$V_{k}(n)=(n−k)!n! n=10k=4V_{4}(10)=(10−4)!10! =6!10! =10⋅9⋅8⋅7=5040$

### The number of variations: 5040

# 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.

## Foundation of combinatorics in word problems

- Flags

How many different flags can be made from green, white, blue, red, orange, yellow, and purple materials, so each flag consists of three different colors? - Toys

3 children pulled 9 different toys from a box. How many ways can toys be divided, so each child has at least one toy? - No. of divisors

How many different divisors have number 13^{ 4}* 2^{ 4}? - Peak

Uphill leads 2 paths and one lift. a) How many options back and forth are there? b) How many options to get there and back by the not same path are there? c) How many options back and forth are there that we go at least once a lift?

- Variations

Find the number of items when the count of variations of the fourth class without repeating is 42 times larger than the count of variations of the third class without repetition. - 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? - Three-digit numbers

How many three-digit numbers are from the numbers 0 2 4 6 8 (with/without repetition)? - Variations 3rd class

From how many elements can we create 13,800 variations of the 3rd class without repeating? - Dices throws

What is the probability that the two throws of the dice: a) Six falls even once b) Six will fall at least once

- Metals

In the Hockey World Cup, play eight teams, and determine how many ways they can win gold, silver, and bronze medals. - Two-digit 3456

Write all the two-digit numbers that can be composed of the digit 7,8,9 without repeating the digits. Which ones are divisible b) two, c) three d) six? - Four-digit 3912

Create all four-digit numbers from digits 1,2,3,4,5, which can repeat. How many are there? - Variation equation

Solve combinatorics equation: V(2, x+8)=72 V(2,x+8) is variations, second class, from x+8 items. - Three reds

What is the probability that all three of the seven cards will be red if three cards are drawn?

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