1. Determine how many sets $X$ satisfy both of the following conditions:

- $X \backslash\{1,2,3,4,5\}=\{6,7,8\}$;
- $X \cup\{4,5,6,7,8,9,10\}=\{1,2,3,4,5,6,7,8,9,10\}$.
1. From the second condition, it follows that $X$ is a subset of the set $\{1,2,3,4,5,6,7,8,9,10\}$. From the first condition, it follows that it must contain the numbers 6, 7, and 8, and must not contain the numbers 9 and 10. From the second condition, it follows that it must contain the numbers 1, 2, and 3. Therefore, the numbers 4 and 5 remain, for which we can arbitrarily choose whether they should be in the set $X$ or not. Thus, there are four such sets: $\{1,2,3,4,5,6,7,8\},\{1,2,3,5,6,7,8\},\{1,2,3,4,6,7,8\}$ and $\{1,2,3,6,7,8\}$.