Calculate Number of Non-Onto Functions Between Sets

To find the number of non-onto functions $f : \{1, 2, 3, 4\} \rightarrow \{a, b, c\}$, we subtract the number of onto functions from the total number of functions.

1. The total number of possible functions from a domain of size $4$ to a codomain of size $3$ is $3^4 = 81$.

2. The number of onto functions is determined using the Principle of Inclusion-Exclusion: $\text{Number of onto functions} = 3^4 - \binom{3}{1} 2^4 + \binom{3}{2} 1^4 = 81 - 48 + 3 = 36$

3. Subtracting the number of onto functions from the total number of functions gives the number of non-onto functions: $\text{Number of non-onto functions} = 81 - 36 = 45$

Thus, the correct answer is 45, which corresponds to Option B.