JEE MAIN - Mathematics (2025 - 7th April Morning Shift - No. 24)
For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________
Answer
13
Explanation
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 $.
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