Explanation:

To perform calculations and store data, a computer fundamentally uses the binary number system. This is because computers are electronic devices that operate using electrical signals, which are typically in one of two states: on or off. These two states are represented by the binary digits (bits) $0$ and $1$. All data storage, processing, and calculations within a computer are performed using this binary system, where the CPU's transistors act as switches, representing these binary states. \nOption (a) Decimal: The decimal number system (base $10$) is what humans commonly use, with digits $0-9$. While computers interact with users in decimal (e.g., displaying numbers), their internal operations are not in decimal. \nOption (b) Hexadecimal: The hexadecimal number system (base $16$) uses digits $0-9$ and letters $A-F$. It is often used by programmers as a compact shorthand to represent binary data because it's easily convertible, but it's not the system computers use for fundamental internal operations. \nOption (c) Octal: The octal number system (base $8$) uses digits $0-7$. Like hexadecimal, it was historically used as a compact representation of binary data, but it is not the internal operating system of a computer. \nOption (e) None of these: This is incorrect as Binary is the correct answer.