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.