Combinations and Permutations

Factorial

The product of all positive integers less than or equal to a given integer n, denoted as n!.

n factorial

The value obtained by multiplying all positive integers from n down to 1, represented as n!.

Combinations

The selection of items where the order does not matter, calculated using the formula C(n, r) = n! / (r! * (n - r)!).

Permutations

The arrangement of items where the order matters, calculated using the formula P(n, r) = n! / (n - r)!.

C(n, r)

Denotes the number of combinations of n items taken r at a time.

P(n, r)

Denotes the number of permutations of n items taken r at a time.

Order of selection

Refers to the importance of the sequence in which items are arranged or chosen, crucial for permutations.

20 combination 3

The number of ways to choose 3 students from 20, equal to 1,140.

10 permutation 5

The number of ways to arrange 5 items from a group of 10, equal to 30,240.

Replacement in combinations

In combinations, items are not replaced once chosen, hence the order does not matter.

Replacement in permutations

In permutations, items are not replaced once selected, but the order is significant.