# Permutations without repetition

The calculator calculates the number of permutations of n elements. Number of permutations is the number of ways to choose a sample of n elements from a set of n distinct objects where order does matter and repetition are not allowed. There are n! (n factorial) ways of arranging n objects into an ordered sequence.## 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

## 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?

## Foundation of combinatorics in word problems

- A pizza

A pizza place offers 14 different toppings. How many different three-topping pizzas can you order? - 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? - Socks

Ben's favorite colors are blue and green. He has six blue socks and six green socks in his sock drawer. Unfortunately, they are completely mixed up, and one day, he has to grab some socks to wear in complete darkness. How many socks (minimum) does he have - A fair coin

A fair coin is tossed twice. Write down the set of possible outcomes. What is the probability of obtaining it? I. Exactly one head ii. No head - Logik game

Letter game Logik is a two-player game that has the following rules: 1. The first player thinks five-letter word in which no letter is not repeated. 2. The second player writes a five-letter word. 3. The first player answers two numbers. The first number - Different 66944

It was Tibor's birthday, and he bought 8 different cookies for his friends (Horalky, Tatanky, Kávenky, Attack, Mila, Anita, Mäta, Lina). He put them all in a box, and each friend could choose two pieces. Tanya chose first. Which two cookies could Táňa cho - Five couples

In how many ways can 5 couples arrange themselves in a row if they stay together? - 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? - Three workplaces

How many ways can we divide nine workers into three workplaces if they need four workers in the first workplace, 3 in the second workplace, and 2 in the third? - School trip

The class has 19 students. How can students be accommodated in the hostel, where available 3× 2-bed, 3× 3-bed and 1× 4-bed rooms? (Each room has its unique number) - Kenneth 2

Kenneth has 100 pennies, 20 nickels, 10 dimes, and 4 quarters. How many ways can he choose coins that total 25 cents? - A basket 4

A basket contains 9 fruits, where 4 are oranges, and the rest are mangoes. Three fruits are taken out one at a time and put aside. Find the probability that 3 are oranges. - 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? - 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? - Constructed 67424

There are six lines 3 cm, 4 cm, 5 cm, 7 cm, 8 cm, and 9 cm long, two of each length. How many isosceles triangles can be constructed from them? List all options.

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