counting formula

Permutations and combinations are mathematical concepts used to count the various ways objects from a set can be selected. A permutation considers the order of selection, while a combination does not. These concepts are fundamental to combinatorics, the field of mathematics that deals with problems of selection and arrangement within finite systems.12

The number of permutations of n objects taken k at a time is calculated using the formula:
[ P(n, k) = \frac{n!}{(n-k)!} ]
The number of combinations of n objects taken k at a time is calculated using the formula:
[ C(n, k) = \frac{n!}{k!(n-k)!} ]
These are known as counting formulas because they help determine the number of possible arrangements or selections without listing them all. Early developments in combinatorics can be traced back to ancient China and India, with significant contributions in the 17th century from mathematicians like Blaise Pascal and Pierre de Fermat, who explored these ideas in relation to probability theory.13