<p>If \(A = \{1, 2, 3, 4\}\) and \(f: A \to A\), then total number of invertible functions \(f\) such that \(f(2) \neq 2\), \(f(4) \neq 4\), \(f(1) = 1\) is equal to:</p>
Step-by-Step Solution
Key Concept: Since f is invertible, it must be a bijection (one-to-one and onto). With f(1)=1 fixed and f(2)≠2, f(4)≠4, we need to count permutations of {2,3,4} that map to {2,3,4} while satisfying the constraints.
<p><strong>Step 1:</strong> Understand invertibility. Since f: A → A is invertible, f must be a bijection (permutation of A). For a finite set, invertible ⟺ bijective ⟺ one-to-one and onto.</p><p><strong>Step 2:</strong> Apply the fixed constraint. We're given f(1) = 1 is fixed. Therefore, f acts as a permutation on the remaining set {2, 3, 4}.</p><p><strong>Step 3:</strong> Count valid permutations of {2,3,4}. We need f restricted to {2,3,4} such that f(2) ≠ 2 and f(4) ≠ 4. Let's enumerate all permutations of {2,3,4} and check which ones satisfy these constraints.</p><p><strong>Step 4:</strong> List all 3! = 6 permutations of {2,3,4}:</p><p> (i) 2→2, 3→3, 4→4: Violates f(2)≠2 and f(4)≠4 ✗</p><p> (ii) 2→2, 3→4, 4→3: Violates f(2)≠2 ✗</p><p> (iii) 2→3, 3→2, 4→4: Violates f(4)≠4 ✗</p><p> (iv) 2→3, 3→4, 4→2: f(2)=3≠2 ✓ and f(4)=2≠4 ✓</p><p> (v) 2→4, 3→2, 4→3: f(2)=4≠2 ✓ and f(4)=3≠4 ✓</p><p> (vi) 2→4, 3→3, 4→2: f(2)=4≠2 ✓ and f(4)=2≠4 ✓</p><p><strong>Step 5:</strong> Count valid functions. From the above enumeration, permutations (iv), (v), and (vi) satisfy both constraints f(2)≠2 and f(4)≠4.</p><p><strong>Step 6:</strong> Each valid permutation of {2,3,4} gives one invertible function f: A → A with f(1)=1, f(2)≠2, and f(4)≠4.</p><p><strong>∴ Answer:</strong> C (3 invertible functions)</p>
Correct Answer: C