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 (integer)
180
Solution
Total number of onto functions
<br/><br/>$$
\begin{aligned}
& =\frac{5 !}{3 ! 2 !} \times 4 ! \\\\
& =\frac{5 \times 4}{2} \times 24=240
\end{aligned}
$$
<br/><br/>When $f(a)=1$, number of onto functions
<br/><br/>$$
\begin{aligned}
& =4 !+\frac{4 !}{2 ! 2 !} \times 3 ! \\\\
& =24+36=60
\end{aligned}
$$
<br/><br/>So, required number of onto functions <br/><br/>$=240-60=180$
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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