Final answer:
The effective user id (EUID) can be set using the permission number 4744, where '4' sets the set-user-ID (setuid) on an executable file, while 777 and 2666 do not affect the EUID.
Step-by-step explanation:
An effective user id (EUID) can be set in a Unix-like operating system to allow a program to run with the privileges of a different user, and this is often used by programs that need to perform tasks that require higher privileges than those granted to the calling user. The permission 4744 is related to setting the set-user-ID (setuid) on an executable file. The first digit '4' is what sets the setuid bit. A permission like 777 or 2666 does not set the setuid bit; '7' and '6' would respectively grant all permissions and both read and write permissions to the user, group, and others, without altering the EUID of the process that runs the executable. Therefore, the correct answer is 4744, where the '4' specifies the setuid bit and the following '744' grants the owner all permissions (read, write, execute), and the group and others read permissions plus execute permissions for the owner.