Tricky Problem: How Many States Can a 5-Bit Binary Counter Represent?

Correctly apply 2^n for total state count, distinguishing it from the counter's maximum numeric value (2^n - 1), a frequent off-by-one confusion.

How many states can a 5-bit binary counter represent? Use the formula 2^5.

Why This Problem Is Tricky

The two related quantities — 'how many states exist' and 'what is the highest value the counter reaches' — differ by exactly one, because the counting sequence starts at 0. This one-off difference makes it very easy to answer the wrong question: computing 2^5 - 1 = 31 answers 'what is the maximum count value,' not 'how many total states are there,' which is genuinely 2^5 = 32 (the count of values 0 through 31 inclusive).

Step-by-Step Worked Solution

A binary counter with n bits has 2^n possible bit patterns, since each of the n bits can independently be 0 or 1: 2 x 2 x 2 x 2 x 2 = 32 for n=5.

These 32 unique bit patterns represent the decimal values 0 through 31 — 32 distinct values in total, because the sequence includes 0 as its own state (it is not skipped or excluded).

So a 5-bit binary counter represents 32 states (00000 through 11111 in binary, or 0 through 31 in decimal).

Why the Obvious Wrong Answer Is Wrong

The common wrong answer is '31 states,' computed as 2^5 - 1 = 31. That formula correctly gives the counter's HIGHEST reachable value (its maximum count), but it answers a different question than 'how many states exist.' Because the count sequence begins at 0 rather than 1, the total number of distinct states (32: the values 0 through 31) is always one MORE than the highest value in that sequence (31). Conflating 'maximum value' with 'number of states' is the specific off-by-one trap this problem targets.

Code Examples

The two formulas are computed side by side and explicitly labeled with what each one actually measures, to make clear that 2^n (total states) and 2^n - 1 (maximum value) answer different questions and are not interchangeable. Expected result: 32

A 5-bit binary counter represents 32 states.

n = 5 bits

Total states       = 2^n = 2^5 = 32   (the values 0 through 31, inclusive)
Maximum count value = 2^n - 1 = 31    (a DIFFERENT quantity: the largest single value)

32 states != 31 states -- these are two different numbers describing two different things.

5-Bit Counter Value Range

00000  00001  00010  ...  11110  11111
   0      1      2    ...   30     31
 |------------------- 32 total states -------------------|
                                        ^ highest value = 31 (= 32 - 1)

Why This Off-by-One Matters in Practice

This exact confusion appears constantly in real hardware and software design: an n-bit address bus addresses 2^n distinct memory locations (not 2^n - 1), a loop meant to visit every state of an n-bit counter must run 2^n times, and a counter's designed overflow/rollover point must be set at 2^n - 1 -> 0, not one value earlier or later. Getting this off-by-one wrong leaves one memory location unreachable or causes a counter to roll over a cycle early or late.

31 states (computing 2^5 - 1) 32 states (computing 2^5) 2^5 - 1 = 31 answers 'what is the highest value this counter can reach,' not 'how many total states does the counter have.' Because the sequence starts at 0, there are 2^5 = 32 distinct values (0 through 31 inclusive), one more than the maximum value itself.

How many states can a 5-bit binary counter represent? A. 31 B. 32 C. 25 D. 16

B. 32 — Total states = 2^n = 2^5 = 32, covering the values 0 through 31 inclusive. 31 (option 1) results from computing 2^5 - 1, which gives the counter's highest reachable VALUE, not its total number of states — those are two different quantities that differ by exactly one because counting starts at 0.

An n-bit counter's highest representable value equals its total state count minus what? A. 0 B. 1 C. n D. 2

B. 1 — Total states = 2^n; highest value = 2^n - 1. So the highest value is always exactly one less than the total state count, because the counting sequence includes 0 as its first state, shifting every subsequent value's position down by one relative to a 1-indexed count.

How many states can a 7-bit binary counter represent, and what is the value of its highest state?

States = 2^7 = 128. Highest value = 2^7 - 1 = 127.

Apply 2^n for the state count and 2^n - 1 for the highest value, separately.