Question

Let \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{a, b, c\} \). Then the number of functions which are not onto are:

• A. 84
• B. 93
• C. 100
• D. 54

Hint:

The number of "not onto" functions is simply the sum of the terms we subtract in the inclusion-exclusion formula for onto functions. In this case, it is \( ^3C_1(2^5) - ^3C_2(1^5) = 93 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the number of functions that are not onto, we first calculate the total number of possible functions from set \( A \) to set \( B \) and then subtract the number of functions that are onto (surjective). An onto function is one where every element in the codomain \( B \) has at least one pre-image in the domain \( A \).
Step 2: Key Formula or Approach:
Total functions from \( A \) (size \( n \)) to \( B \) (size \( m \)) = \( m^n \).
Number of onto functions = \( m^n - ^mC_1(m-1)^n + ^mC_2(m-2)^n - \dots \)
Here, \( n = 5 \) and \( m = 3 \).
Step 3: Detailed Explanation:
Total number of functions = \( 3^5 = 243 \).
Now, calculate the number of onto functions using the principle of inclusion-exclusion: \[ \text{Onto Functions} = 3^5 - [^3C_1(3-1)^5 - ^3C_2(3-2)^5] \] \[ = 243 - [3(2^5) - 3(1^5)] \] \[ = 243 - [3(32) - 3] \] \[ = 243 - [96 - 3] = 243 - 93 = 150 \] The number of functions that are not onto is: \[ \text{Total functions} - \text{Onto functions} = 243 - 150 = 93 \]
Step 4: Final Answer:
The number of functions which are not onto is 93.