Combinations C(n,k)

 The COMBINATIONS (_nC_k) calculator computes the number of combination possible in a set of k elements containing exactly one of each unique element from a finite set of n objects where order does not matter. 

INSTRUCTIONS: Enter the following:

• (n) Total Number of Objects  (e.g. 52 cards).
• (k) Number in Subset  (e.g. 5 card poker hand)

Combinations (_nC_k): The calculator computes and return the number of combinations.   Combinations also plays the core role in the Binomial Coefficient and in the construction of Pascal's Triangle.

The Math / Science

The combination equation computes the number of possible combinations of k elements (unique sets) from within the set of n objects.  Unlike permutations, the order of the distinct selected items are not considered in the set of combinations. We denote the number of combinations as C(n,k).  The number C(n,k) or `C_"(n,k)"` is the the probability of choosing one specific set of k elements from a set of n-objects.

The formula for the number of combinations is as follows:

          _nC_k =  `(n!)/(k!(n-k)!)`

where:

• nCk = combinations of k objects from a set of n objects
• n = total size of set
• k = size of subset

EXAMPLES

This equations can be used to answer a question like:  How many different hands can be dealt in a 5 card Poker deal. 

The answer is:  52! / ( 5! ( 52-5)! ) = 2598960

So, from a set of 52 cards, there are 2,598,960 possible different hands that could be dealt.

Another example would be the number of combinations of players on the field from a 50 player football team where every player can play any position on both offense or defense.  Since each player is a unique person and the number of players on the field is eleven, the number of possible player combinations would be:

C(50,11) = 50! /  ( 11! (50-11)! ) = 37,353,738,800

Combinatorics Calculators

• n! - factorial
• `n!  = sqrt(2 pi n) (n/e)^n` - Stirling's factorial approximation
• _nC_k - combinations
• C_r(n,k) - combinations with repetitions
• _nP_k - permutations
• P_r(n) - permutations with repetitions
• P_comp(n,2) - Comparison permutations
• P_c(n,r) - Circular Permutation