Question

a panel containing three on-off switches in a row is to be set. assuming no restrictions, construct a tree diagram to list all the possible panel settings. select the correct tree diagram where \on\ and \off\ are labeled 1 and 0, respectively. \\(\bigcirc\\) a. first switch, second switch, third switch, switch settings (tree diagram structure as shown) \\(\bigcirc\\) b. first switch, second switch, third switch, switch settings (tree diagram structure as shown) \\(\bigcirc\\) c. first switch, second switch, third switch, switch settings (tree diagram structure as shown) \\(\bigcirc\\) d. first switch, second switch, third switch, switch settings (tree diagram structure as shown)

Explanation:

Step1: Analyze the first switch

The first switch can be either 0 (off) or 1 (on). For each choice of the first switch, we consider the second switch, and then the third.

Step2: Analyze the second switch for first switch = 0

If the first switch is 0, the second switch can be 0 or 1.

• If second switch is 0, third switch can be 0 or 1, giving settings 000, 001.
• If second switch is 1, third switch can be 0 or 1, giving settings 010, 011.

Step3: Analyze the second switch for first switch = 1

If the first switch is 1, the second switch can be 0 or 1.

• If second switch is 0, third switch can be 0 or 1, giving settings 100, 101.
• If second switch is 1, third switch can be 0 or 1, giving settings 110, 111.

Now we check the options:

• Option A: The first switch has two 0s, which is incorrect (first switch should have 0 and 1, each once).
• Option B: Let's check the settings. For first=0, second=0: third=1 gives 001 (correct), second=1: third=0 gives 010, third=1 gives 011. For first=1, second=0: third=1 gives 101, second=1: third=0 gives 110, third=1 gives 111. Wait, but when first=1, second=0, third=0 should be 100, but in option B, first=1, second=0, third=1 is 101, but where is 100? Wait no, let's re - check. Wait in option B, for first=1, second=0: third=1 (101), second=1: third=0 (110), third=1 (111). But when first=1, second=0, third=0 is missing? Wait no, maybe I misread. Wait no, let's check option B again. Wait the first switch branches into 0 and 1. For first=0: second=0 branches to third=1 (001), second=1 branches to third=0 (010) and third=1 (011). For first=1: second=0 branches to third=1 (101), second=1 branches to third=0 (110) and third=1 (111). Wait, but the possible settings for three switches (0/1) are 000, 001, 010, 011, 100, 101, 110, 111. Option B is missing 000 and 100? Wait no, maybe I made a mistake. Wait option D: Let's check. First switch 0: second 0: third 0 (000), third 1 (001); second 1: third 0 (010), third 1 (011). First switch 1: second 0: third 0 (100), third 1 (101); second 1: third 0 (110), third 1 (111). Yes, option D has all 8 possible settings (since each switch has 2 states, 2^3 = 8 settings: 000, 001, 010, 011, 100, 101, 110, 111). Let's check option B again. In option B, for first=0, second=0, third=1 (001); second=1, third=0 (010), third=1 (011). For first=1, second=0, third=1 (101); second=1, third=0 (110), third=1 (111). So 000 and 100 are missing. Option C: For first=0, second=0, third=0 (000), third=1 (001); second=0, third=0 (000), third=1 (001) – duplicate, and first=1, second=0, third=0 (100), third=1 (101); second=0, third=0 (100), third=1 (101) – duplicate. Option A: First switch has two 0s, which is wrong. So option D is correct.

Answer:

D. First Switch, Second Switch, Third Switch, Switch Settings with the correct branching as described (000, 001, 010, 011, 100, 101, 110, 111)