JEE Main 2025 Jan 22 Shift 2, Mathematics Q13: Types of Functions

Total functions $A\to B$: $4^4=256$. Bijections (one-one with $|A|=|B|=4$): $4!=24$. So many-one functions in total: $256-24=232$.

Among these many-one functions, we want those for which $1\in f(A)$, i.e. the image contains $1$.

Count many-one functions with $1\notin f(A)$: image is contained in $B\setminus \{1\}=\{4,9,16\}$, so each of the 4 inputs maps to one of 3 outputs, giving $3^4=81$ functions. Every such function is automatically many-one (pigeon-hole: 4 inputs into 3 outputs).

Hence many-one functions with $1\in f(A)$ equal $232-81=151$.