Let $\mathbb{N}$ denote the set of all positive integers. Consider the sets $$\mathrm{A}=\{1,2,3,4,5\} \text { and } \mathrm{B}=\{1,2,3,4,5,6,7\} .$$ Let S be the set of all functions $f: \mathrm{A} \rightarrow \mathrm{B}$ such that $f(2) \neq 2$ and $f(4) \neq 4$. Consider the set $\mathrm{T}=\left\{f \in \mathrm{~S}:\right.$ there exists a function $\mathrm{g}: \mathrm{B} \rightarrow \mathbb{N}$ such that $\mathrm{g}(f(\mathrm{x}))=2^{\mathrm{x}}$ for all $\left.\mathrm{x} \in \mathrm{A}\right\}$. Then the number of elements in the set T is ____.

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Relations And Functions XII

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 \to B

such that

f (2) \neq = 2

and

f (4) \neq = 4

. Consider the set

T = {f \in S :

there exists a function

g : B \to N

such that

g (f (x)) = 2^{x}

for all

x \in A}

.
Then the number of elements in the set T is ____.

Answer: 1860

Since

g (f (x)) = 2^{x}

for all

x \in A

f

must be one-one. Total one-one functions from A to B:

^{7} C_{5} \times 5! = 2520

. Functions with

f (2) = 2

^{6} C_{4} \times 4! = 360

. Functions with

f (4) = 4

^{6} C_{4} \times 4! = 360

. Functions with both

f (2) = 2

and

f (4) = 4

^{5} C_{3} \times 3! = 60

. By inclusion-exclusion:

∣ T ∣ = 2520 - 360 - 360 + 60 = 1860

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Numerical answer