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 $n(S_5)$ is equal to

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Permutations and Combinations

For

n \geq 2

, let

S_{n}

denote the set of all subsets of

{1, 2, \dots, n}

with no two consecutive numbers. For example

{1, 3, 5} \in S_{6}

, but

{1, 2, 4} \in / S_{6}

. Then

n (S_{5})

is equal to

Answer: 13

Counting the subsets of

{1, 2, 3, 4, 5}

that contain no two consecutive elements by size, the number with exactly

r

chosen elements is

(r n - r + 1)

with

n = 5

(1 5) = 5

of size one,

(2 4) = 6

of size two,

(3 3) = 1

of size three, together with the empty set. These add to

1 + 5 + 6 + 1 = 13

, so

n (S_{5}) = 13

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Integer answer