How can we visualize that $2^n$ gives the number of ways binary digits of length n?

Question

Like if we have to find the number of ways can be represented in bits up to 4 places. We use $2^4$, but why do we use this method?

Answer 1

A bit has values $1$ or $0$, so $2$ values. You can prove by induction. Suppose that you have $k$ bits and you have $2^k$ ways and you add one more bit. Then you will have that initial $2^k$ ways for every value of the new bit, so $2*2^k=2^{k+1}$

Answer 2

Hint:

Note that in a binary sequence of length $n$ because it is binary we have the option of choosing either a $0$ or a $1$ to fit each of those $n$ places.

What are the number of sequences, then?

Answer 3

The literal meaning of the positional numeral system is that digit in the $n$th place (counting from zero) represents $r^n$, where $r$ is the radix or base. For example, in decimal, $1000 = 10^3$ because there are $3$ zeros. There are one thousand numbers below $1000$, counting zero.

Similarly in binary, there are $2^n$ numbers representable with $n$ bits.

Answer 4

Because for each bit you have 2 different simbols.

Then if $n$ denotes the number of bits, this means that we have to make a choice between $1$ or $0$ for $n$ times is: $$\underbrace{2\cdot2\cdot2\cdot\dots\cdot2}_{n\text{-times}}=2^n$$ This happens because every choice is independent from the others.

If you want you can prove it easily by induction.

Answer 5

To visualize the method, I would use a binary tree : for each digit you have two choices $0$ and $1$ or left and right in the tree. At the end you have $2^n$ possible ways to get to the last line of the tree.