Question

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6). Then the number of functions f: A→B satisfying f(1) + f(2) = f(4)-1 is equal to _________ .

Answer 1

Correct Answer: 360

We are tasked with finding the total number of functions f that satisfy the given conditions. 

Step 1: Analyze Cases for f(1) and f(2)

The sum f(1) + f(2) must satisfy f(1) + f(2) &leq 5, and both f(1) and f(2) are integers. Let's explore the possible values for f(1) and f(2):

• Case (i): f(1) = 1

If f(1) = 1, then f(2) can take values from 1, 2, 3, 4 (4 possible mappings).

• Case (ii): f(1) = 2

If f(1) = 2, then f(2) can take values from 1, 2, 3 (3 possible mappings).

• Case (iii): f(1) = 3

If f(1) = 3, then f(2) can take values from 1, 2 (2 possible mappings).

• Case (iv): f(1) = 4

If f(1) = 4, then f(2) can only take the value 1 (1 possible mapping).

Step 2: Count Mappings for f(5) and f(6)

Both f(5) and f(6) can each take any of 6 possible values independently.

Step 3: Calculate the Total Number of Functions

To compute the total number of functions, we calculate the number of ways to choose f(1), f(2), f(5), and f(6):

• For f(1) and f(2), we sum the possibilities from the cases: 
  (4 + 3 + 2 + 1) = 10
• For f(5) and f(6), each has 6 possible mappings: 
  6 × 6 = 36

Thus, the total number of functions is: 
10 × 36 = 360

Final Answer:

The total number of functions is 360.