Question:

<p><strong>Step 1: Interpret the condition</strong></p><p>f(f(x)) = f(x) for all x ∈ A means f(y) = y for all y in the range of f.</p><p>If y = f(x) for some x, then f(y) = y (y is a fixed point).</p><p><strong>Step 2: Structure of such functions</strong></p><p>Choose a non-empty subset S ⊆ A to be the set of fixed points (range of f).</p><p>For each x ∈ A: if x ∈ S, then f(x) = x; if x ∉ S, then f(x) can be any element of S.</p><p><strong>Step 3: Count functions for each subset S</strong></p><p>For a fixed subset S with |S| = k (where k ≥ 1):</p><p>• Elements in S: must map to themselves (1 choice each)</p><p>• Elements not in S: can map to any element in S (k choices each)</p><p>• Number of such functions = k^(5-k)</p><p><strong>Step 4: Sum over all possible subsets</strong></p><p>Total = Σ(k=1 to 5) C(5,k) · k^(5-k)</p><p>= C(5,1)·1^4 + C(5,2)·2^3 + C(5,3)·3^2 + C(5,4)·4^1 + C(5,5)·5^0</p><p>= 5(1) + 10(8) + 10(9) + 5(4) + 1(1)</p><p>= 5 + 80 + 90 + 20 + 1</p><p>= <strong>196</strong></p>