No Two Consecutive Elements? Count Subsets FAST ⚡
❓ Question For n ≥ 2 n \ge 2 , let S n S_n denote the set of all subsets of { 1 , 2 , … , n } \{1,2,\dots,n\} with no two consecutive numbers . For example: { 1 , 3 , 5 } ∈ S 6 , { 1 , 2 , 4 } ∉ S 6 . \{1,3,5\}\in S_6,\quad \{1,2,4\}\notin S_6. Then the value of n ( S 5 ) n(S_5) (i.e. the number of elements in S 5 S_5 ) is equal to ? 🖼️ Question Image ✍️ Short Solution This is a classic JEE combinatorics + recurrence question. Key idea: 👉 Subsets with no consecutive integers follow the Fibonacci pattern. We’ll first understand why , then apply it to n = 5 n=5 . 🔹 Step 1 — Understand the restriction We are forming subsets of { 1 , 2 , 3 , 4 , 5 } \{1,2,3,4,5\} such that: You may include a number But if you include k k , you cannot include k + 1 k+1 So every choice blocks the next number. 🔹 Step 2 — Define a counting approach Let: f ( n ) = number of subsets of { 1 , 2 , … , n } with no consecutive...