Formula Sheet: Permutations and Combinations

n! (factorial) represents the product of all positive integers up to n. Useful for calculating permutations.

nPr denotes the number of permutations of n objects taken r at a time, where order matters.

nCr denotes the number of combinations of n objects taken r at a time, where order does not matter.

This is an alternative form to calculate the number of permutations without using factorials when order matters.

This identity establishes that choosing r objects from n is equivalent to choosing (n - r) objects from n.

It represents the number of ways to choose no objects, highlighting that there is exactly one way to choose nothing.

This shows that there are n ways to choose 1 object from n total objects.

This is Pascal's identity and shows how combinations can be related recursively.

This states that the number of permutations of n objects taken all at once is n!.

This equation relates the number of permutations to combinations by dividing the permutations by r!, accounting for the rearrangements of r objects.

Provides the formula for calculating the total different arrangements of n distinct objects taken r at a time.

Calculates how many ways n distinct items can be chosen without considering the order.

Refers to how many ways we can arrange r items when we can select the same item more than once.

Useful in scenarios involving combinations of objects with repetitions allowed.

Another form of combinatorial identity which expresses combinations in terms of factorials.

This specific case of combinations calculates the number of ways to choose 2 items from n.

Describes how to arrange p people in q specific seats.

Where p1, p2,..., pk are the frequencies of indistinguishable letters in the word.

This formula calculates combinations specifically for choosing 2 objects from n.

Expresses the valid combinations when items are not all distinct.