Permutations and Combinations

States that if one event can occur in m ways and a second in n ways, the two events together can occur in m times n ways.

Gives the total number of linear arrangements of n distinct objects taken all at a time.

Gives the number of ordered arrangements of n distinct objects taken r at a time without repetition.

Gives the number of arrangements when selecting r items from n distinct items with repetition allowed.

Gives the number of arrangements of n objects where p_1 are of one kind, p_2 of another kind, and so on.

Gives the number of circular arrangements of n distinct objects when clockwise and anticlockwise orders are distinguished.

Gives the number of unordered selections of r objects from n distinct objects.

Relates permutations to combinations by ordering the r selected objects in r! ways.

Expresses complementary selection and the condition under which two combinations with the same upper index are equal.

Combines two combinations of order n to form a single combination of order n+1.

Gives the number of non-negative integer solutions to a sum equation or distributing n identical items into r bins.

Gives the number of distinct straight lines formed by connecting n points where m points are collinear.

Gives the highest exponent of prime p dividing n!.