Question

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

• A. 48
• B. 45
• C. 51
• D. 35

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the number of "not onto" functions, we first calculate the total number of possible functions and then subtract the number of "onto" (surjective) functions.

Step 2: Key Formula or Approach:
1. Total functions from set $A$ (size $n$) to set $B$ (size $m$) is $m^n$. 2. Number of onto functions from size $n$ to size $m$ is given by: \[ m^n - \binom{m}{1}(m-1)^n + \binom{m}{2}(m-2)^n - \dots \]

Step 3: Detailed Explanation:
1. Total functions from $\{1, 2, 3, 4\}$ to $\{a, b, c\}$: \[ 3^4 = 81 \] 2. Number of onto functions: - Ways to pick the codomain: $m=3$, domain $n=4$. - Onto functions = \( 3^4 - \binom{3}{1}2^4 + \binom{3}{2}1^4 \) - Onto functions = \( 81 - 3(16) + 3(1) = 81 - 48 + 3 = 36 \). 3. Number of functions that are NOT onto: \[ \text{Total} - \text{Onto} = 81 - 36 = 45 \]

Step 4: Final Answer:
The number of functions which are not onto is 45.