### $Onto function :\space$ $x$ $\longrightarrow$ $y$

### $x\space is\space preimage\space of\space y\space ,\space y\space is\space image\space of\space x\space$

$\textbf {Step 1:}\space Find\space non\space onto\space function\space is\space relatively\space easy\space than\space find\space onto\space function\space so\space -\space$

$non\space onto\space function\space =\space Either\space a\space not\space mapped\space OR\space B\space not\space mapped\space OR\space C\space not\space mapped\space$

${So\space we\space need\space to\space find\space}$ $|$$S_a$ $\cup$ $S_b$ $\cup$ $S_c$$|$ $=\space ?$

$\textbf{Step 2:}\space According\space to\space \textbf{Principal of Mutual Inclusion and Exclusion}$

$|$ $S_a$ $\cup$ $S_b$ $|$ = $|$ $S_a$ $|$ $+$ $|$ $S_b$ $|$ $-$ $|$ $S_a$ $\cap$ $S_b$ $|$

$|$ $S_a$ $|$ $=$ $It$ $means\space a\space not\space mapped$ $=$ $1^n$

$|$ $S_b$ $|$ $=$ $It$ $means\space b\space not\space mapped$ $=$ $1^n$

$if\space both\space not\space mapped$ $=$ $0^n$

$|$ $S_a$ $\cup$ $S_b$ $|$ = $1\space +\space 1\space -\space 0\space =\space 2$

$\textbf{Step 3:}\space So\space Onto\space functions\space =\space Total\space functions\space -\space non\space onto\space functions $

$And\space as\space we\space know\space total\space functions\space =\space 2^n$

$So\space onto\space functions\space are\space =\space 2^n - 2$

$Hence\space Option\space C\space is\space right\space one$