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

Total onto functions from $$R = \{a,b,c,d,e\}$$ to $$S = \{1,2,3,4\}$$:

Using inclusion-exclusion: $$4^5 - \binom{4}{1} \cdot 3^5 + \binom{4}{2} \cdot 2^5 - \binom{4}{3} \cdot 1^5$$

$$= 1024 - 4(243) + 6(32) - 4(1) = 1024 - 972 + 192 - 4 = 240$$

Now subtract onto functions where $$f(a) = 1$$. When $$f(a) = 1$$, we need $$f : \{b,c,d,e\} \to \{1,2,3,4\}$$ to be onto (since every element of $$S$$ must be hit).

But wait — element 1 is already hit by $$f(a) = 1$$. So $$\{b,c,d,e\}$$ must cover $$\{2,3,4\}$$ (all of them), and may also map to 1.

Number of such functions = total functions from 4 elements to 4 elements that cover $$\{2,3,4\}$$:

$$= 4^4 - \binom{3}{1} \cdot 3^4 + \binom{3}{2} \cdot 2^4 - \binom{3}{3} \cdot 1^4$$

$$= 256 - 3(81) + 3(16) - 1 = 256 - 243 + 48 - 1 = 60$$

Onto functions with $$f(a) \neq 1$$ = $$240 - 60 = 180$$