Variations without repetition

The calculator computes 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.

Elements to choose from:

(n)

Elements chosen:

(k)

Calculate

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 the order of the group's elements matters.

The number of variations can be easily calculated using the combinatorial rule of product. For example, if we have a set of n = 5 numbers {1, 2, 3, 4, 5} and we need to make third-class variations, then V(3,5) = 5 × 4 × 3 = 60.

$V_k(n)=n(n-1)(n-2)...(n-k+1)=\frac{n!}{(n-k)!}$

n! is called the factorial of n, which is the product of the first n natural numbers. The factorial notation is clearer and equivalent. For calculations, it is sufficient to use the procedure derived from the combinatorial rule of product.

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.

more math problems »