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.

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 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 V₃ (5) = 5 * 4 * 3 = 60.

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

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

• Seating
  How many ways can 7 people sit on 5 numbered chairs (e.g., seat reservation on the train)?
• 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?
• Olympics
  How many ways can six athletes be placed on the podium at the Olympics? Depends on the color of the metal.
• Permutations 6450
  Seven times the permutations of n elements equal one-eighth of the permutations of n + 2 elements. What is the number of elements?
• Three-digit 6690
  How many three-digit numbers do we make from the numbers 4,5,6,7?
• Five-digit numbers
  How many different five-digit numbers can be created from the number 2,3,5 if the number 2 appears in the number twice and the number 5 also twice?
• Refrigerator, lemonades
  How many possible ways can we store three lemonades, four mineral waters, and two juices in the refrigerator next to each other?
• Playmakers + coach
  In a basketball game, two pivots, two wings, and one point guard play. The coach has three pivots, four wing players, and two playmakers available on the bench. How many different five players can a coach send to the board during a game?
• Password dalibor
  Kamila wants to change the password daliborZ by a) two consonants exchanged between themselves, b) changes one little vowel to such same great vowel c) makes these two changes. How many opportunities do you have a choice?
• Together 3331
  The group has 12 red girls and 25 blue girls in costumes. How many of them can we put together a group of 6 girls so that the four girls have red outfits?
• Practice
  How many ways can you place 20 pupils in a row when starting on practice?
• Year 2020
  The four-digit number divided by 2020 gives a result of 1, **. (Can not be in form 1,*0. ) Write all the options.
• Coffe cups
  We have 4 cups with four different patterns. How many possible combinations can we create from 4 cups?
• Variations 26791
  If the number of elements increases by two, the number of variations of the second class of these elements created by 38 increases. What is the original number of elements?
• You have
  You have four reindeer and want to have 3 fly your sleigh. You always have your reindeer fly in a single-file line. How many different ways can you arrange your reindeer?

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