Option D, 13 combinations.
Step-by-step explanation:
We have 5 switches, let's write this as:
_ _ _ _ _
Now, we need to divide in cases.
1) We do not have any switch in the off position.
Here we have only one arrangement:
ON ON ON ON ON
2) There is one switch in the OFF position.
In this case, we can never have two switches in the off position next to each other (because there is only one)
in this case, we will have 5 permutations (The OFF one can be any of the 5 switches)
3) There are two switches in the OFF position.
Now, let's suppose that we have the first switch in the OFF position:
OFF _ _ _ _
The one just at the right can not be the other OFF one, but the other 3 can be, then we have 3 combinations.
Now let's suppose that the second switch is in the OFF position:
_ OFF _ _ _
the only other positions where we can have the other OFF switch is in the fourth and fifth position, then here we have 2 combinations.
Now suppose that the switch is the third position:
_ _ OFF _ _
There are two positions where the other OFF switch can be, first and fifth one. But we already counted the case where we have the first and third switch in the OFF position, so in this case we have only one combination left.
And the same happens if we fix the fourth switch in the OFF position, we already counted all the combinations here.
In total we have 3 + 2 + 1 = 6 combinations with two switches in the OFF position.
4) 3 switches in the OFF position.
Here we only have one combination, this is:
OFF ON OFF ON OFF
And we can not have 4 or 5 switches in the OFF position, because in those cases we would have adjacent switches in the OFF position.
Now we need to add all the combinations we found.
0 switches in the OFF position: 1 combination
1 switch in the OFF position: 5 combinations
2 switches in the OFF position: 6 combinations
3 switches in the OFF position: 1 combination:
Total combinations = 1 + 5 + 6 + 1 = 13
The correct option is D: 13
