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QJEEM23Ma22c7: JEE Main 2023 Mathematics - Class 11 JEEM23Ma22c7 Solution: Let and . Total number of onto functions such that , is equal…

Let R={a,b,c,d,e}\mathrm{R}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\} and S={1,2,3,4}\mathrm{S}=\{1,2,3,4\} . Total number of onto functions f:R→Sf: \mathrm{R} \rightarrow \mathrm{S} such that f(a)≠1f(\mathrm{a}) \neq 1 , is equal to ______________.

Step-by-Step Verified Solution

Total number of onto functions

amp;=5!3!2!×4!amp;=5×42×24=240\begin{aligned}& =\frac{5 !}{3 ! 2 !} \times 4 ! \\\\& =\frac{5 \times 4}{2} \times 24=240\end{aligned}

When f(a)=1f(a)=1 , number of onto functions

amp;=4!+4!2!2!×3!amp;=24+36=60\begin{aligned}& =4 !+\frac{4 !}{2 ! 2 !} \times 3 ! \\\\& =24+36=60\end{aligned}

So, required number of onto functions

=240−60=180=240-60=180

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