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JEE Advanced2026MathematicsFunctionsActual

Let N denote the set of all positive integers. Consider the sets A = 1, 2, 3, 4, 5 and B = 1, 2, 3, 4, 5, 6, 7 . Let S be the set of all functions f: A B such that f(2) 2 and f(4) 4 . Consider the set T = f S : there exists a function g : B N such that g(f(x)) = 2^x for all x A . Then the number of elements in the set T is ___________.

Correct answer

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Step-by-step solution

The condition g(f(x)) = 2^x for all x A implies that if f(x₁) = f(x₂) , then g(f(x₁)) = g(f(x₂)) , which gives 2^ x₁ = 2^ x₂ x₁ = x₂ . Thus, f must be an injective (one-to-one) function. Conversely, if f is injective, such a function g can always be constructed. Therefore, T is the set of all injective functions f: A B such that f(2) 2 and f(4) 4 . The total number of injective functions from A to B is ⁷P₅ = 7 6 5 4 3 = 2520 . Let E₂ be the set of injective functions where f(2) = 2 . The number of such functions is

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