JEE Main 2025 — Sets And Relations Question with Solution

To find $n(S_5)$, which is the number of subsets of $\{1,2,3,4,5\}$ with no consecutive numbers, we start by enumerating these subsets.

Let's denote the set $\{1,2,3,4,5\}$ as $A$. The subsets of $A$ that meet the criteria are:

The empty set: $\{\}$

Single-element sets: $\{1\}$, $\{2\}$, $\{3\}$, $\{4\}$, $\{5\}$

Two-element sets with no consecutive numbers: $\{1,3\}$, $\{1,4\}$, $\{1,5\}$, $\{2,4\}$, $\{2,5\}$, $\{3,5\}$

Three-element set with no consecutive numbers: $\{1,3,5\}$

Counting these subsets, we have:

1 subset with zero elements

5 subsets with one element

6 subsets with two elements

1 subset with three elements

Adding these counts, there are $1+5+6+1=13$ subsets in total.

Thus, $n(S_5)=13$.