Let $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$ and $\mathrm{B}=\{1,2,3,4\}$. Then the number of elements in the set $\mathrm{C}=\{\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B} \mid 2 \in \mathrm{f}(\mathrm{A})$ and $\mathrm{f}$ is not one-one $\}$ is
$\mathrm{C}=\{\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B} \mid 2 \in \mathrm{f}(\mathrm{A})$ and $\mathrm{f}$ is not one-one $\}$
Case-I : If $\mathrm{f}(\mathrm{x})=2 \forall \mathrm{x} \in \mathrm{A}$ then number of function $=1$
Case-II : If $f(x)=2$ for exactly two elements then total number of many-one function $={ }^{3} \mathrm{C}_{2}{ }^{3} \mathrm{C}_{1}=9$
Case-III : If $\mathrm{f}(\mathrm{x})=2$ for exactly one element then total number of many-one functions $={ }^{3} \mathrm{C}_{1}{ }^{3} \mathrm{C}_{1}=9$
Total $=19$
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