Question

How many subsets does the set {1, 2, ..., n} have that contain no two consecutive integers if 1 and n also count as consecutive?

Answer 1

The number of subsets containing no two consecutive integers from the set {1, 2, ..., n} (where 1 and n are considered consecutive) is 2^(n-1).

Step-by-step explanation:

Consider the set {1, 2, ..., n} and its subsets. To count the subsets without consecutive integers, observe that for each element, it can either be included or excluded from a subset. If we include a number, its consecutive number cannot be included. Hence, for each element, there are two choices: either include it or exclude it to avoid consecutive integers.

For the first element, there are two options: include 1 or exclude 1. For the second element, regardless of whether 1 is included or excluded, there's only one choice as 2 cannot follow 1. Similarly, for each subsequent element up to n, there's only one choice as each number cannot be consecutive with its preceding element.

This results in a total of 2 choices for each of the (n-1) elements following the first one. Hence, the total number of subsets with no consecutive integers is calculated as 2^(n-1), as each element's inclusion or exclusion represents a binary decision, resulting in 2 possibilities raised to the power of (n-1) elements after the first one.