JEE Advanced 2026 · Paper 2 · Mathematics Q10

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MathematicsNumerical+4Question 10 of 54

Let $\mathbb N$ denote the set of all positive integers. Consider the sets
$A=\{1,2,3,4,5\}\quad\text{and}\quad B=\{1,2,3,4,5,6,7\}.$
Let $S$ be the set of all functions $f:A\to B$ such that $f(2)\ne2$ and $f(4)\ne4$ . Consider the set
$T=\left\{f\in S:\text{there exists a function }g:B\to\mathbb N\text{ such that }g(f(x))=2^x\text{ for all }x\in A\right\}.$
Then the number of elements in the set $T$ is ________.

Your answer

Answer and worked solution

Official answer: 1860

Condition on $f$ . If $f(x_1)=f(x_2)$ then $2^{x_1}=g(f(x_1))=g(f(x_2))=2^{x_2}$ , so $x_1=x_2$ : $f$ must be one-one. Conversely, for one-one $f$ define $g$ by $g(f(x))=2^x$ (and anything in $\mathbb N$ elsewhere). So $T$ is the set of one-one $f$ with $f(2)\ne2$ , $f(4)\ne4$ .

Count by inclusion-exclusion:

all one-one maps $A\to B$ : $7\cdot6\cdot5\cdot4\cdot3=2520$ ;
with $f(2)=2$ : $6\cdot5\cdot4\cdot3=360$ ; with $f(4)=4$ : $360$ ;
with both: $5\cdot4\cdot3=60$ .

$|T|=2520-720+60=1860.$

Topics:Types of functions Permutations

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JEE Advanced 2026 · Paper 2, Mathematics question 10