Variations with repetition

The calculator computes the number of variations with repetition. A variation of the k-th class with repetition of n elements is any ordered k-element group composed of only these n elements such that each element can be repeated any number of times.

Calculation:

V_{k}^{'} (n) = n^{k} n = 10 k = 4 V_{4}^{'} (10) = 1 0^{4} = 10000

Number of variations with repetition: 10000

A bit of theory - the foundation of combinatorics

Variations with repetition

A variation with repetition of the k-th class of n elements is an ordered k-element group formed from a set of n elements, where elements can be repeated and order matters. A typical example is forming numbers from the digits 2, 3, 4, 5 and counting how many such numbers exist. We calculate the count using the combinatorial rule of product:

V_{k}^{'} (n) = n \cdot n \cdot n \cdot n . . . n = n^{k}

Foundation of combinatorics in word problems

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)
Flags
How many different flags can be made from green, white, blue, red, orange, yellow, and purple materials if each flag consists of three stripes of different colours?
Toys
3 children pulled 6 different toys from a box. How many ways can toys be divided so each child has at least one toy?
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?
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
Eight teams play in the Hockey World Cup. Determine how many ways gold, silver, and bronze medals can be awarded.
Choosing Colors for Sweater
Danka knits a sweater and has a choice of seven colors. a) How many ways can he choose three colors for the sleeves? b) He wants ribbons of four colors on his back. How many options does he have to choose from?
Three-digit number
Find all three-digit numbers n with three different non-zero digits divisible by the sum of all three two-digit numbers we get when we delete one digit in the original number.
Darts Score Combinations
Adam was practicing for a darts competition in class. Every day at home, he threw darts at a target in which the individual fields were worth 1,3 and 5 points. He threw 9 darts every day and always scored 27 points. He is in good form and never missed a t
A pizza
A pizza place offers 14 different toppings. How many different three-topping pizzas can you order?
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?
Dance couple arrangement
Six boys and six girls (among them Emil, Felix, Gertrude, and Hannah) want to dance. The number of ways they can make six (mixed) couples if Emil does not want to dance with Gertrude and Hannah wants to dance with Felix is?
Five-digit number repetition
How many five-digit numbers can we make from digits 2,3,4,6,7,9 if they can repeat with the digits?
Cookie selection ways
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
Isosceles triangle construction
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.