For $\mathrm{n} \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2 \ldots . . ., n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in \mathrm{S}_6$, but $\{1,2,4\}
otin \mathrm{S}_6$. Then $n\left(\mathrm{~S}_5 ight)$ is equal to ________

Question

For $\mathrm{n} \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2 \ldots . . ., n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in \mathrm{S}_6$, but $\{1,2,4\} 
otin \mathrm{S}_6$. Then $n\left(\mathrm{~S}_5ight)$ is equal to ________

Aicharya · Accepted Answer

$\mathrm{A}=\{1,2,3,4,5 \ldots . . . \mathrm{n}\}$
No. of subsets having $r$ elements such that no two are consecutive is $={ }^{n-r+1} C_r$
for $\mathrm{n}=5$, no. of ways $={ }^{6 \cdot \mathrm{r}} \mathrm{C}_{\mathrm{r}}$
Subsets having no element $=1$
Subsets having exactly 1 element $={ }^5 \mathrm{C}_1=5$
Subsets having exactly 2 element $={ }^4 \mathrm{C}_2=6$
Subsets having exactly 3 element $={ }^3 \mathrm{C}_3=1$
$\Rightarrow 5+6+1+1=13$

For $\mathrm{n} \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2 \ldots . . ., n\}$ with no two…

Solution