A collection of non-empty subsets of a set whose union is the original set.

A division of a set into disjoint subsets.

A collection of subsets of a set whose intersection is empty.

A collection of subsets of a set whose union is the original set and whose intersection is empty.

(n!)

(2^n)

(n^n)

(n)

(\frac{e^{-x}}{x})

(\frac{1}{1-x})

(\frac{1}{1+x})

(\frac{1}{1-x^2})

The number of ways to partition a set of (n) elements into (k) non-empty subsets.

The number of ways to choose (k) elements from a set of (n) elements.

The number of ways to arrange (n) elements in a row.

The number of ways to divide a set of (n) elements into two non-empty subsets.

(B(n) = \sum_{k=1}^n S(n, k))

(B(n) = \prod_{k=1}^n S(n, k))

(B(n) = \sum_{k=0}^n S(n, k))

(B(n) = \prod_{k=0}^n S(n, k))

(\frac{1}{(1-x)^n})

(\frac{1}{(1+x)^n})

(\frac{1}{(1-x^2)^n})

(\frac{1}{(1+x^2)^n})

A method for counting the number of elements in the union of two or more sets.

A method for counting the number of elements in the intersection of two or more sets.

A method for counting the number of elements in the symmetric difference of two or more sets.

A method for counting the number of elements in the complement of a set.

(|\cup_{i=1}^n A_i| = \sum_{i=1}^n |A_i| - \sum_{i

(|\cup_{i=1}^n A_i| = \sum_{i=1}^n |A_i| + \sum_{i

(|\cup_{i=1}^n A_i| = \sum_{i=1}^n |A_i| - \sum_{i

(|\cup_{i=1}^n A_i| = \sum_{i=1}^n |A_i| + \sum_{i

To count the number of ways to arrange objects.

To count the number of ways to select objects from a set.

To count the number of ways to distribute objects into groups.

All of the above.

To calculate the probability of an event.

To calculate the expected value of a random variable.

To calculate the variance of a random variable.

All of the above.

To design efficient algorithms.

To analyze the complexity of algorithms.

To design data structures.

All of the above.

To study the properties of matter.

To study the properties of energy.

To study the properties of space-time.

All of the above.

To study the properties of markets.

To study the properties of firms.

To study the properties of consumers.

All of the above.

To study the properties of cells.

To study the properties of organisms.

To study the properties of ecosystems.

All of the above.