Let R = a, b, c, d, e and S = 1, 2, 3, 4 . Total number of onto...

Given,

$R = { a, b, c, d, e } and S = { 1, 2, 3, 4 }$

Now taking, $f ( a ) = 1$ we get, one of $f ( b ) , f ( c ) , f ( d ) , f ( e ) = 1$ then total such cases $= 4 \cdot 3 ! = 24$
Now if, only $f ( a ) = 1$, then we have distribute \left\{2, 3, 4\right\}$amongst \left\{b, c, d, e\right\}$,
So, total cases $= {3}^{4} - \left(C1 3 \cdot {2}^{4}\right) + \left(C2 3 \cdot 1\right)<br />$= 36$<br />So, number of onto functions when$f ( a ) = 1$is$24 + 36 = 60$</p><p>Now finding, total number of onto functions,<br />$= {4}^{5} - \left(C1 4 \cdot {3}^{5}\right)$+ \left(C2 4 \cdot {2}^{5}\right) - \left(C3 4 \cdot 1\right)<br />$= 1024 - 973 + 192 - 4$<br />$= 240$<br />Number of required functions when$f\left(a\right)$\ne 1$ will be,

$= 240 - 60= 180$