To solve this problem, we will use Lucas' Theorem to determine the sequence of $\binom{n}{k} \mod 2$ for $k = 0, 1, \ldots, 7$, where $n = 1152921504606846987$.

First, we need to express $n$ in binary: $n = 1152921504606846987$

Convert this number to binary: $n = 10000000000000000000000000000000000000000000000000000000000011$

We need to compute $\binom{n}{k} \mod 2$ for $k = 0, 1, \ldots, 7$. Lucas' Theorem states that $\binom{n}{k} \mod 2$ is determined by comparing the binary representation of $n$ and $k$. Specifically, $\binom{n}{k} \mod 2 = 1$ if and only if for every bit position $i$, the bit of $k$ is less than or equal to the bit of $n$.

We will compare the binary representation of $n$ with each $k$ in the range $0, 1, \ldots, 7$:

1. For $k = 0$ $k = 00000000$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{0} \mod 2 = 1$.

2. For $k = 1$ $k = 00000001$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{1} \mod 2 = 1$.

3. For $k = 2$ $k = 00000010$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{2} \mod 2 = 1$.

4. For $k = 3$ $k = 00000011$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{3} \mod 2 = 1$.

5. For $k = 4$ $k = 00000100$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{4} \mod 2 = 1$.

6. For $k = 5$ $k = 00000101$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{5} \mod 2 = 1$.

7. For $k = 6$ $k = 00000110$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{6} \mod 2 = 1$.

8. For $k = 7$ $k = 00000111$ Since all bits of $k$ are less than or equal to the corresponding bits of $n$, $\binom{n}{7} \mod 2 = 1$.

Hence, the sequence of $\binom{n}{k} \mod 2$ for $k = 0, 1, \ldots, 7$ is: $11111111$

The answer as a simple 0/1 string is: $\boxed{11111111}$

Do you want more details or have any questions?

Here are 5 numbered questions you can ask:

1. Can you explain Lucas' Theorem in more detail?
2. How do you convert a decimal number to binary?
3. Can you demonstrate Lucas' Theorem with another example?
4. What is the significance of the modulus operation in combinatorics?
5. Can you explain the concept of binomial coefficients and their properties?

Tip: When working with large numbers, converting them to binary can simplify the analysis, especially when using theorems like Lucas' Theorem.