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

• Seven-segmet
  Helen is amused that he punched a calculator (seven-segment display) number and used only digits 2 to 9. Some numbers have the property that She again gave their image in the axial or central symmetry some number. Determine the maximum number of three-dig
• Metals
  Eight teams play in the Hockey World Cup. Determine how many ways gold, silver, and bronze medals can be awarded.
• Four swords
  Obelix has three helmets, four swords, and five shields. How many words must you make at the blacksmith forge Metallurgix to be able to walk another 90 days in unique armor?
• Alarm clock
  The old watchmaker has a unique digital alarm in its collection that rings whenever the sum of the alarm's digits equals 21. Find out when the alarm clock will ring. What is their number? List all options.
• Word MATEMATIKA
  How many words can be created from the phrase MATEMATIKA by changing the letters' order, regardless of whether the words are meaningful?
• Cinema seating
  Seven boys are sitting next to each other in the cinema. How many ways can they sit on the seats if the boys want to sit next to each other?
• Couple train boarding
  Ten married couples board the train, which has five cars. How many ways can they take if no two spouses want to be in the exact vehicle?
• Element arrangement ways
  Determine the number of arrangements of these six elements: a, a, a, b, b, c.
• Three digit from four digits
  How many three-digit numbers can you make using the digits 4,6,7 and 9?
• Probability of picking
  There are 5 chocolate, 3 cottage cheese, and 2 apricot croissants in the bag. Croissants are randomly selected in bags. What is the probability of drawing 1 chocolate, 1 cheese, and 1 apricot croissant without returning?
• Wagon assembly ways
  How many ways can we assemble five wagons when sand is in three wagons and cement in two?
• Shirt short combinations
  Dustin has 8 T-shirts and three pairs of shorts in his closet. How many ways can he dress for school?
• Seven segments display
  Electronic devices sometimes use the type of digits below, where each digit uses some short stripes. For example, seven uses three small stripes. What is the largest three-digit number that you can make if you use twenty stripes?
• Probability - coin
  When tossing a coin 4 times in a row, what is the probability that a sign will come up 3 times?
• Relay
  A relay race will be run by the class team of Kate, Alice, Michaela, and Erica. Determine how many different running orders are possible, given that each of them can run in any position.

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