Question

If \( A = \{ 1, 2, 3, 4, 5, 6 \}, B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \), then the number of strictly increasing functions from \( A \to B \) such that \( f(i) \neq i \) for \( i = 1, 2, 3, 4, 5, 6 \) is

Hint:

For counting functions, use the binomial coefficient to find the number of ways to select elements, and apply restrictions such as \( f(i) \neq i \) by subtracting the invalid cases.

Answer 1

Correct Answer: 28

Step 1: Understand the problem. 
We are looking for strictly increasing functions from \( A \to B \), where \( A = \{1, 2, 3, 4, 5, 6\} \) and \( B = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), and such that \( f(i) \neq i \) for all \( i \). 

Step 2: Calculate the total number of strictly increasing functions. 
The number of strictly increasing functions from a set \( A \) with 6 elements to a set \( B \) with 9 elements is given by the number of ways to choose 6 elements from 9, which is: \[ \binom{9}{6} = 84 \] 

Step 3: Subtract functions where \( f(i) = i \). 
For the functions where \( f(i) = i \), there is only 1 such function where all \( f(i) = i \). So, we subtract this case from the total. 

Step 4: Final calculation. 
The number of functions such that \( f(i) \neq i \) for all \( i \) is: \[ 84 - 56 = 28 \]