n choose k calculator

Find out how many different ways you can choose k items from n items set without repetition and without order. This number is also called combination number or n choose k or binomial coefficient or simply combinations. See also general combinatorial calculator.

Elements to choose from:

(n)

Elements chosen:

(k)

Calculate

A bit of theory - the foundation of combinatorics

Combinations

A combination of a k-th class of n elements is an unordered k-element group formed from a set of n elements. The elements are not repeated, and it does not matter the order of the group's elements. In mathematics, disordered groups are called sets and subsets. Their number is a combination number and is calculated as follows:

$C_k(n)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$

A typical example of combinations is that we have 15 students and we have to choose three. How many will there be?

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)?
• Insurance
  The house owner is insured against natural disasters and pays 0.05% annually of the value of the house 88 Eur. Calculate the value of the house. Calculate the probability of disaster if you know that 50% of the insurance is to pay damages.
• Football league
  In the 5th football league is 10 teams. How many ways can be filled first, second, and third place?
• Tournament
  Determine how many ways can be chosen two representatives from 34 students to school tournament.

• Sales
  From statistics of sales goods, item A buys 57% of people, and item B buys 76% of people. What is the probability that from 18 people buy 10 item A and 8 item B?
• Event probability
  The probability of event N in 5 independent experiments is 0.4. What is the probability that the event N occurs in one experiment (chance is the same)?
• Rectangles
  How many rectangles with area 8855 cm² whose sides are natural numbers?
• Rectangle
  In a rectangle with sides, 8 and 9 mark the diagonal. What is the probability that a randomly selected point within the rectangle is closer to the diagonal than any side of the rectangle?
• 7 heroes
  6 heroes galloping on 6 horses behind. How many ways can we sort them behind?

• Win in raffle
  The raffle tickets were sold to 200, 5 of which were winning. What is the probability that Peter, who bought one ticket, will win?
• Pairs
  At the table sit 10 people, 5 on one side and 5 on the other side. Among them are 3 pairs. Every pair wants to sit opposite each other. How many ways can they sit?
• Count of triangles
  On each side of an ABCD square is 10 internal points. Determine the number of triangles with vertices at these points.
• Cars plates
  How many different license plates can a country have since they use 3 letters followed by 3 digits?
• 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?

• Ace
  We pulled out one card from a complete set of playing cards (32 cards). What is the probability of pulling the ace?

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