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The problem asks:

"Aaliyah’s computer printer allows for optional settings with a panel of five on-off switches in a row. How many different settings can she select if no two adjacent switches can both be on?"

### Key Constraints:
- There are 5 on-off switches.
- No two adjacent switches can be on at the same time.

### Approach:

We need to determine how many valid configurations of the switches exist, where no two adjacent switches are both "on." Each switch can either be in the "on" or "off" position.

### Step-by-step solution:

Let’s define the problem as choosing switches to be "on" while making sure no two adjacent switches are "on."

1. **Recursion Relation**: 
   - This problem is equivalent to finding how many ways you can arrange a sequence where:
     - Switch \( S_1 \) can either be "on" or "off."
     - If \( S_1 \) is "on," then \( S_2 \) must be "off" (to prevent adjacent switches from both being on).
     - If \( S_1 \) is "off," then \( S_2 \) can either be "on" or "off."
   - This structure is similar to a recursive relation where each configuration depends on the previous two switches.

   Let \( f(n) \) be the number of valid configurations for \( n \) switches.
   - For \( n = 1 \), you can have the following configurations: "on" or "off." So, \( f(1) = 2 \).
   - For \( n = 2 \), you can have these configurations:
     - Both switches "off" (valid).
     - First switch "on," second "off" (valid).
     - First switch "off," second "on" (valid).
     Therefore, \( f(2) = 3 \).
   - For \( n = 3 \), you either turn on the first switch (forcing the second off) or leave the first switch off. Therefore, the number of valid configurations follows the recursive relation:
     \[
     f(n) = f(n-1) + f(n-2)
     \]
   - This is similar to the Fibonacci sequence starting at \( f(1) = 2 \), \( f(2) = 3 \).

### Solution for 5 switches:

To find \( f(5) \):
\[
f(3) = f(2) + f(1) = 3 + 2 = 5
\]
\[
f(4) = f(3) + f(2) = 5 + 3 = 8
\]
\[
f(5) = f(4) + f(3) = 8 + 5 = 13
\]

So, there are **13 different valid configurations** of the 5 switches, where no two adjacent switches are both "on."

Let me know if you need more details or have any questions!

### Further related questions:
1. What is the formula to compute the number of valid configurations for \( n \) switches?
2. How does this problem relate to the Fibonacci sequence?
3. How would the solution change if the number of switches was increased to 6 or 7?
4. How many valid configurations exist if exactly 3 switches must be on, with no two adjacent?
5. How can dynamic programming be applied to this type of problem?

### Tip:
Whenever solving problems involving restrictions on adjacent elements (like switches or seating), consider using recursion or dynamic programming for an efficient solution.

Aaliyah’s computer printer allows for optional settings with a panel of five on-off switches in a row. How many different settings can she select if no two adjacent switches can both be on?

Explore how to determine the number of valid configurations for five on-off switches where no two adjacent switches can be on at the same time. This problem involves combinatorics and recursion, relating to a Fibonacci-like sequence. Perfect for students in Grades 9-12.

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