The number of functions $$f: \{1,2,3,4\} \to \{a,b,c\}$$, which are not onto, is :

The set in the domain has $$4$$ elements and the set in the co-domain has $$3$$ elements.

Step 1: Count all possible functions.
For every element of the domain we may choose any of the $$3$$ images independently, so the total number of functions is
$$3^{4}=81$$

Step 2: Subtract the onto (surjective) functions.
A function $$f:\{1,2,3,4\}\to\{a,b,c\}$$ is onto if each of $$a,b,c$$ actually appears as an image.
We count surjective functions by the Principle of Inclusion-Exclusion (PIE).

• Start with all functions: $$3^{4}$$.
• Subtract those that miss at least one specific element of the co-domain.
  Choosing which one is missed: $$\binom{3}{1}$$ ways.
  If that element is excluded, only $$2$$ images remain, giving $$2^{4}$$ functions.
  Hence this term is $$\binom{3}{1}\,2^{4}=3\cdot16=48$$.
• Add back the functions that miss two specific elements (they were subtracted twice).
  Choosing the two missed: $$\binom{3}{2}$$ ways.
  If two are excluded, only $$1$$ image remains, giving $$1^{4}=1$$ function.
  Hence this term is $$\binom{3}{2}\,1^{4}=3\cdot1=3$$.

Therefore, the number of onto functions is
$$3^{4}-\binom{3}{1}2^{4}+\binom{3}{2}1^{4}=81-48+3=36$$

Step 3: Count the non-onto functions.
Non-onto functions = Total functions − Onto functions:
$$81-36=45$$

Thus, the required number of functions that are not onto is $$45$$.

Option B which is: $$45$$