How many subsets does a set with "n" elements have?

How many subsets does a set with "n" elements have?

• A. 2^n
• B. n^2
• C. n!
• D. n+1

Answer: (A)

A set with "n" elements has \(2^n\) subsets due to the fundamental principle of combinations. Each element in the set has two possibilities: it can either be included in a subset or excluded from it. Consequently, for each of the n elements, the choices of inclusion or exclusion multiply together. To illustrate, consider a set with a small number of elements, such as a set with 3 elements, say {a, b, c}. Each element can be included in a subset or not included, leading to the following combinations: - Include none: {} - Include a: {a} - Include b: {b} - Include c: {c} - Include a and b: {a, b} - Include a and c: {a, c} - Include b and c: {b, c} - Include all: {a, b, c} In total, there are 8 subsets, which is \(2^3\) since \(3\) is the number of elements. This pattern holds for any set size, thus confirming that a set with "n" elements indeed has \(2^n\) subsets. This principle is foundational in combinatorics and helps in various areas

A set with "n" elements has (2^n) subsets due to the fundamental principle of combinations. Each element in the set has two possibilities: it can either be included in a subset or excluded from it. Consequently, for each of the n elements, the choices of inclusion or exclusion multiply together.

To illustrate, consider a set with a small number of elements, such as a set with 3 elements, say {a, b, c}. Each element can be included in a subset or not included, leading to the following combinations:

• Include none: {}

• Include a: {a}

• Include b: {b}

• Include c: {c}

• Include a and b: {a, b}

• Include a and c: {a, c}

• Include b and c: {b, c}

• Include all: {a, b, c}

In total, there are 8 subsets, which is (2^3) since (3) is the number of elements. This pattern holds for any set size, thus confirming that a set with "n" elements indeed has (2^n) subsets. This principle is foundational in combinatorics and helps in various areas