JEE MAIN - Mathematics (2023 - 8th April Evening Shift - No. 17)
Let $$\mathrm{R}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$$ and $$\mathrm{S}=\{1,2,3,4\}$$. Total number of onto functions $$f: \mathrm{R} \rightarrow \mathrm{S}$$
such that $$f(\mathrm{a}) \neq 1$$, is equal to ______________.
Answer
180
Explanation
Total number of onto functions
$$ \begin{aligned} & =\frac{5 !}{3 ! 2 !} \times 4 ! \\\\ & =\frac{5 \times 4}{2} \times 24=240 \end{aligned} $$
When $f(a)=1$, number of onto functions
$$ \begin{aligned} & =4 !+\frac{4 !}{2 ! 2 !} \times 3 ! \\\\ & =24+36=60 \end{aligned} $$
So, required number of onto functions
$=240-60=180$
$$ \begin{aligned} & =\frac{5 !}{3 ! 2 !} \times 4 ! \\\\ & =\frac{5 \times 4}{2} \times 24=240 \end{aligned} $$
When $f(a)=1$, number of onto functions
$$ \begin{aligned} & =4 !+\frac{4 !}{2 ! 2 !} \times 3 ! \\\\ & =24+36=60 \end{aligned} $$
So, required number of onto functions
$=240-60=180$
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