### Instruction: Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$?### Response: I need to find all the positive integers that have four digits in base $2$ and add them up.
The smallest four-digit number in base $2$ is $1000_2$, which is equivalent to $8_{10}$ in base $10$.
The largest four-digit number in base $2$ is $1111_2$, which is equivalent to $15_{10}$ in base $10$.
So I need to find the sum of all the numbers from $8$ to $15$ in base $10$ and then convert that to base $2$.
I can use the formula for the sum of an arithmetic series to find the sum of the numbers from $8$ to $15$ in base $10$.
The formula is $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the nth term.
In this case, $n = 15 - 8 + 1 = 8$, $a_1 = 8$, and $a_n = 15$.
So the sum of the numbers from $8$ to $15$ in base $10$ is $S_8 = \frac{8}{2}(8 + 15) = 4 \times 23 = 92$.
Now I need to convert $92$ from base $10$ to base $2$.
I can do this by repeatedly dividing by $2$ and writing down the remainders.
$92 \div 2 = 46$ with a remainder of $0$.
$46 \div 2 = 23$ with a remainder of $0$.
$23 \div 2 = 11$ with a remainder of $1$.
$11 \div 2 = 5$ with a remainder of $1$.
$5 \div 2 = 2$ with a remainder of $1$.
$2 \div 2 = 1$ with a remainder of $0$.
$1 \div 2 = 0$ with a remainder of $1$.
So the base $2$ representation of $92$ is obtained by reading the remainders from bottom to top: $1011100_2$.
This is the final answer.