# Permutations, k-Permutations and Combinations

Permutations, k-Permutations and Combinations

Number of Permutations

The number of ways to arrange different objects in a row is . The exclamation mark "!" is read as "factorial". Of course, the product is the same in reverse order: . Each such arrangement is called a permutation. For consistency, it is assumed that .

n

n !=1×2×3×⋯×

n

n !=n×(n-1)×(n-2)×…3×2×1

0 !=1

Number of k-Permutations

If only of the objects are to be arranged in a row, the formula is

k

n

P(n,k)==n×(n-1)×…(n-k+1)

n !

(n-k) !

with factors. If , . Such an arrangement is called a partial permutation, or a -permutation. Clearly , because all objects are being arranged; the formula reduces to because the denominator is .

k

k>n

P(n,k)=0

k

P(n,n)=n !

n

n !

(n-n) !=0 !=1

Number of Combinations

The number of ways to choose a subset of objects from objects is

k

n

C(n,k)=

n !

k !(n-k) !

Therefore, . Each choice of a subset is called a combination. Another notation for is . Again, if , . A special case is .

C(n,k)=P(n,k)/k !

C(n,k)

n |

k |

k>n

C(n,k)=0

C(n,n)=1