Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $f:…

Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $f: \mathrm{A} \rightarrow \mathrm{B}$ such that $1 \in f(\mathrm{~A})$ is equal to :
  1. 151
  2. 139
  3. 163
  4. 127

Solution

$\begin{aligned}
& A=\{1,2,3,4\} \\ & B=\{1,4,9,16\}
\end{aligned}$
Total number of functions $=4^4$
Total number of one-one functions $=4!$
Total number of many one functions $=4^4-4!=232$
Total number of many-one functions in which $1 \notin f(A)=3 \times 3 \times 3 \times 3=81$
$\therefore$ Total number of many one functions $1 \notin f(A)$
$\begin{aligned}
& =232-81 \\ & =151
\end{aligned}$

Asked in: JEE Main 2025 (22 Jan Shift 2)

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