The question in Swedish asks: "How many binary numbers less than 256 start and/or end with two ones?"

Solution Breakdown:

1. Binary Numbers Less than 256:
   Any number less than 256 can be represented by at most 8 bits (since $2^8 = 256$). So we are looking at binary numbers with 8 or fewer bits.

2. Two Ones at the Beginning or End:
   We need to find the binary numbers that either start or end with two ones, or both. Let’s consider different cases:

Case 1: Numbers that start with two ones

For numbers less than 256, the 8-bit binary numbers that start with two ones look like:
$11xxxxxx$
This leaves 6 remaining bits, each of which can be either 0 or 1. Therefore, there are $2^6 = 64$ possible binary numbers that start with two ones.

Case 2: Numbers that end with two ones

For 8-bit binary numbers, the ones that end with two ones look like:
$xxxxxx11$
Again, there are 6 bits left that can either be 0 or 1. Therefore, there are $2^6 = 64$ possible binary numbers that end with two ones.

Case 3: Numbers that start and end with two ones

The binary numbers that both start and end with two ones are of the form:
$11xxxx11$
In this case, there are 4 middle bits that can be either 0 or 1. Therefore, there are $2^4 = 16$ possible binary numbers that start and end with two ones.

Final Count: Use Inclusion-Exclusion

To find the total number of numbers that either start or end with two ones (or both), we use the inclusion-exclusion principle:

$\text{Total} = (\text{Numbers that start with two ones}) + (\text{Numbers that end with two ones}) - (\text{Numbers that start and end with two ones})$ $\text{Total} = 64 + 64 - 16 = 112$

So, the answer is 112 binary numbers.

Would you like more details on any step, or do you have further questions?

5 Related Questions:

1. How are binary numbers constructed, and what are their main applications?
2. What is the inclusion-exclusion principle, and how does it apply to other counting problems?
3. What are the properties of numbers represented in binary format?
4. How would the solution change if we consider numbers less than $2^n$ for different values of $n$?
5. How many binary numbers between 0 and 512 start and end with exactly two ones?

Tip:

Binary numbers follow powers of 2, so when dealing with binary ranges, remember that each bit doubles the number of possible combinations.