Question:medium
The number of functions \(f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}\), which are not onto, is:
The number of functions \(f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}\), which are not onto, is:
Updated On: Jun 6, 2026
- 48
- 45
- 51
- 35
Show Solution
The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
A function from set \(A\) to set \(B\) is "onto" (surjective) if every element in \(B\) has at least one pre-image in \(A\).
The number of functions that are "not onto" is the total number of functions minus the number of onto functions.
Step 2: Key Formula or Approach:
Let \(n(A) = m\) and \(n(B) = n\).
Total number of functions = \(n^m\).
The number of onto functions can be found using the Principle of Inclusion-Exclusion (PIE):
\[ \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m \] Here, \(m = 4\) and \(n = 3\).
Step 3: Detailed Explanation:
First, let's find the total number of possible functions:
\[ \text{Total functions} = 3^4 = 81 \] Next, we calculate the number of onto functions:
\[ \text{Onto functions} = \binom{3}{0}3^4 - \binom{3}{1}2^4 + \binom{3}{2}1^4 - \binom{3}{3}0^4 \] \[ = 1(81) - 3(16) + 3(1) - 0 \] \[ = 81 - 48 + 3 = 36 \] The number of functions that are NOT onto is:
\[ \text{Not onto functions} = \text{Total functions} - \text{Onto functions} \] \[ = 81 - 36 = 45 \] {Alternative Approach:}
Functions that are not onto miss at least one element in the codomain.
Using PIE directly for "not onto" functions:
\[ N(\text{not onto}) = \binom{3}{1} 2^4 - \binom{3}{2} 1^4 = 3(16) - 3(1) = 48 - 3 = 45 \] Step 4: Final Answer:
The number of functions which are not onto is 45.
A function from set \(A\) to set \(B\) is "onto" (surjective) if every element in \(B\) has at least one pre-image in \(A\).
The number of functions that are "not onto" is the total number of functions minus the number of onto functions.
Step 2: Key Formula or Approach:
Let \(n(A) = m\) and \(n(B) = n\).
Total number of functions = \(n^m\).
The number of onto functions can be found using the Principle of Inclusion-Exclusion (PIE):
\[ \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m \] Here, \(m = 4\) and \(n = 3\).
Step 3: Detailed Explanation:
First, let's find the total number of possible functions:
\[ \text{Total functions} = 3^4 = 81 \] Next, we calculate the number of onto functions:
\[ \text{Onto functions} = \binom{3}{0}3^4 - \binom{3}{1}2^4 + \binom{3}{2}1^4 - \binom{3}{3}0^4 \] \[ = 1(81) - 3(16) + 3(1) - 0 \] \[ = 81 - 48 + 3 = 36 \] The number of functions that are NOT onto is:
\[ \text{Not onto functions} = \text{Total functions} - \text{Onto functions} \] \[ = 81 - 36 = 45 \] {Alternative Approach:}
Functions that are not onto miss at least one element in the codomain.
Using PIE directly for "not onto" functions:
\[ N(\text{not onto}) = \binom{3}{1} 2^4 - \binom{3}{2} 1^4 = 3(16) - 3(1) = 48 - 3 = 45 \] Step 4: Final Answer:
The number of functions which are not onto is 45.
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