Question:

<p><strong>Step 1:</strong> Understand the constraint. f(f(x)) = f(x) means f(x) is a fixed point of f for all x ∈ {1,2,3,4}. So if f(x) = y, then f(y) = y.</p><p><strong>Step 2:</strong> Identify the range structure. The range of f restricted to {1,2,3,4} must consist entirely of fixed points. Let S ⊆ {1,2,3,4} be the set of fixed points (elements where f(x) = x).</p><p><strong>Step 3:</strong> For each x ∈ {1,2,3,4}: either f(x) = x (x is a fixed point), or f(x) ∈ S where S ≠ ∅. This means S must be non-empty.</p><p><strong>Step 4:</strong> Count by cases based on S = {fixed points in {1,2,3,4}}.</p><p>For a non-empty S ⊆ {1,2,3,4}:</p><ul><li>Each element in S must satisfy f(x) = x (1 choice per element)</li><li>Each element NOT in S must map to some element in S (|S| choices per element)</li><li>Element 5 can map to any element in {1,2,3,4,5} (5 choices)</li></ul><p><strong>Step 5:</strong> Total count = Σ (number of non-empty S ⊆ {1,2,3,4}) × |S|^(4-|S|) × 5</p><p>= 5 × [C(4,1)×1³ + C(4,2)×2² + C(4,3)×3¹ + C(4,4)×4⁰]</p><p>= 5 × [4×1 + 6×4 + 4×3 + 1×1]</p><p>= 5 × [4 + 24 + 12 + 1]</p><p>= 5 × 41</p><p>∴ <strong>Answer: 205</strong></p>