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

• N-gon
  How many diagonals does a convex 30-gon have?
• Seating
  How many ways can 7 people sit on 5 numbered chairs (e.g., seat reservation on the train)?
• 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?
• Rectangles
  How many rectangles with an area of 8855 cm² have sides that are natural numbers?

more math problems »