A traffic light controller uses a 5-bit binary pattern to store the state of its lights.
Explain why this pattern can represent a maximum of 32 different states.
Mark scheme
Award up to two marks for a linked explanation such as:
- Each bit has two possible values, 0 and 1 (1) so the five bits give 2 × 2 × 2 × 2 × 2/2^5 = 32 different combinations/states (1)
- Because each of the five bits can be either 0 or 1 (1) there are 2^5/2 × 2 × 2 × 2 × 2 = 32 different combinations/states (1)
- Each extra bit doubles the number of combinations/states (1) so five bits give 2^5 = 32 states (1)
For both marks, the expansion must follow/associate with the statement.
Do not award a response that only repeats the 32 states given in the question, such as 'it can represent 32 states' or '2^5 = 32', without a reason about each bit having two possible values.
Do not accept 2 × 5 = 10 as the calculation; the number of states doubles with each extra bit.
Do not accept 31/0 to 31 as the number of states; 31 is the largest value five bits can store, not the number of states.
Example answers and how they were marked
There are 32 different states because each of the 5 bits has two possible values, 0 and 1, so the number of combinations is 2 × 2 × 2 × 2 × 2 = 32.
Why: Point that each bit has two possible values, 0 and 1 (1), linked to the calculation 2 × 2 × 2 × 2 × 2 = 32 states (1).
Each bit has two possible values, 0 and 1.
Why: Gives the point that each bit has two possible values (1) but does not link it to 2^5 = 32, so the second mark is not earned.
It can represent 32 states because the highest number it can store is 31.
Why: Only repeats the 32 states from the question and gives the common misconception that 31 is the number of states; there is no reason about each bit having two possible values, so no marks.