Given,

$R=\{a,b,c,d,e\}$ and $S=\{1,2,3,4\}$

Now taking, $f\left(a\right)=1$ we get, one of $f\left(b\right), f\left(c\right), f\left(d\right), f\left(e\right)=1$ then total such cases $=4·3!=24$
Now if, only $f\left(a\right)=1$, then we have distribute $\left\{2,3,4\right\}$ amongst $\left\{b,c,d,e\right\}$,
So, total cases $=3^4-\left({}^{3}C_1·2^4\right)+\left({}^{3}C_2·1\right)$
$=36$
So, number of onto functions when $f\left(a\right)=1$ is $24+36=60$

Now finding, total number of onto functions,
$=4^5-\left({}^{4}C_1·3^5\right)+\left({}^{4}C_2·2^5\right)-\left({}^{4}C_3·1\right)$
$=1024-973+192-4$
$=240$
Number of required functions when $f\left(a\right)\ne 1$ will be,

$=240-60$$=180$