Question

If set A has 5 elements, and set B has 7 elements, then the number of one-one functions that can be defined from A to B is

• A. \( 7^5 - 7 \)
• B. \( 5^7 - 5 \)
• C. \( 5^7 - 7P_5 \)
• D. \( 7^5 - 7P_5 \)

Hint:

To count one-one functions, use permutations to ensure elements in the domain map uniquely to elements in the codomain.

Answer 1

The Correct Option is D

Step 1: Calculating the Total Number of Functions Since every element in set \( A \) (which has 5 elements) can map to any of the 7 elements in \( B \), the total number of functions from \( A \) to \( B \) is: \[ 7^5 \]
Step 2: Removing Non-One-One Mappings A function is injective if no two elements in \( A \) map to the same element in \( B \). The number of one-to-one functions is given by: \[ 7P_5 = \frac{7!}{(7-5)!} = \frac{7!}{2!} \]
Final Answer: Thus, the number of one-to-one functions is: \[ 7^5 - 7P_5 \]