Question

Let \( P \) and \( Q \) be two finite sets having 3 elements each. The total number of mappings from \( P \) to \( Q \) is

• A. 32
• B. 516
• C. 6
• D. 9
• E. 27

Hint:

The total number of mappings from a set \( P \) to set \( Q \) is given by \( |Q|^{|P|} \), where \( |P| \) is the number of elements in set \( P \), and \( |Q| \) is the number of elements in set \( Q \).

Answer 1

Answer: (E)

Step 1: Understand the Problem We are given two finite sets \( P \) and \( Q \), each containing 3 elements. We need to determine the total number of possible mappings (functions) from \( P \) to \( Q \). 
Step 2: Recall the Formula for Number of Mappings If \( P \) has \( m \) elements and \( Q \) has \( n \) elements, the total number of mappings (functions) from \( P \) to \( Q \) is given by: \[ n^m. \] This is because each element of \( P \) has \( n \) choices in \( Q \), and the choices are independent. 
Step 3: Apply the Formula Here, \( P \) and \( Q \) each have 3 elements. Therefore:
- \( m = 3 \) (number of elements in \( P \))
- \( n = 3 \) (number of elements in \( Q \))
The total number of mappings from \( P \) to \( Q \) is: \[ n^m = 3^3 = 27. \] 
Step 4: Verify the Answer The total number of mappings is \( 27 \), which corresponds to option (E). 
Final Answer: The total number of mappings from \( P \) to \( Q \) is: \[ \boxed{27}. \]