The name of a variable in the C programming language is a string that can contain uppercase letters, lowercase letters, digits, or underscores. Further, the first character in the string must be a letter, either uppercase or lowercase, or an underscore. If the name of a variable is determined by its first eight characters, how many different variables can be named in C? (Note that the name of a variable may contain fewer than eight characters.)

There are 212, 133, 167, 002, 880, different variables.

Explanation:

Beacuse there are 26 possible uppercase letters, 26 lowercases letters, 10 digits and 1 underscore, which are 63 possible chacracters in total.

Strings of length 8

1º character: 53 ways (uppercase or lowercase letter or an underscore)

2º character to 8º character: we have 63 ways

By using the product rule: 53.63.63.63.63.63. 63. 63 = 53. 63⁷

In the same way we have:

strings of length 7: 53. 63⁶

strings of length 6: 53.63⁵

strings of length 5: 53. 63⁴

strings of length 4: 53.63³

strings of length 3: 53. 63²

strings of length 2: 53.63

strings of length 1: 53

Strings fo length eight or less calculate by using the sum rule:

53₊ 53.63₊ 53.63² ₊ 53.63³ ₊ 53.63⁴ ₊ 53.63⁵ ₊ 53.63⁶ ₊ 53.63⁷ = 212,133,167,002,880