Combination and Permutation Calculator

Combination and permutation formulas help you count the number of subsets of a certain size that can be created from a larger set. Combination refers to counting subsets in which the order of the elements is irrelevant. Permutation refers to counting subsets in which the order does matter, that is, subsets with the same elements but arranged differently count as different subsets.

Combination Example:

As an example, consider the number of ways to select 2 students out of 5 students. If the students names are A, B, C, D, and E, then there are 10 possible combinations: AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE.

If the total number of objects is n and the size of the subset is m, then the number of combinations is written (n C m) or (ⁿ_m). It is read "n choose m."

Permutation Example:

Now consider the number of ways to select 2 students out of 5 students, but the order in which you pick the students matters, for instance, one student will be the leader and the other will be the assistant. Now there are 20 possible permutations: AB, BA, AC, CA, AD, DA, AE, EA, BC, CB, BD, DB, BE, EB, CD, DC, CE, EC, DE, and ED.

If the total number of objects is n and the size of the subset is m, then the number of permutations is written (n P m).

Combination and Permutation Formulas

(n C m) = n!/(m!*(n-m)!)

(n P m) = n!/(n-m)!

The symbol ! is factorial, that is, for any non-negative integer k, k! = (k)*(k-1)*(k-2)*...*(2)*(1). When k = 0, 0! is set to equal 1. Examples of combination and permutation formulas in action:

(5 C 2) = 5!/(2!*3!)
= (5*4*3*2*1)/(2*1*3*2*1)
= (5*4*3)/(3*2*1)
= 60/6 = 10

(5 P 2) = 5!/2!
= (5*4*3*2*1)/(3*2*1)
= 5*4 = 20

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