Question

a common computer programming rule is that names of variables must be between one and eight characters long. the first character can be any of the 26 letters, while successive characters can be any of the 26 letters or any of the 10 digits. for example, allowable variable names include a, bb, and m3477k. how many different variable names are possible? (ignore the difference between uppercase and lowercase letters.) choose the correct answer below. a. 208 b. 2,095,681,645,538 c. 75,444,539,239,368 d. 2,037,468,286,496

Explanation:

Step1: Calculate one - character variable names

The first character can be any of the 26 letters. So, there are 26 one - character variable names, which is simply 26.

Step2: Calculate two - character variable names

The first character has 26 choices (any letter), and the second character has \(26 + 10=36\) choices (any letter or digit). By the multiplication principle, the number of two - character variable names is \(26\times36\).

Step3: Calculate three - character variable names

The first character has 26 choices, and the second and third characters each have 36 choices. So the number of three - character variable names is \(26\times36\times36 = 26\times36^{2}\).

Step4: Calculate four - character variable names

The number of four - character variable names is \(26\times36^{3}\) (26 choices for the first character and 36 choices for each of the remaining three characters).

Step5: Calculate five - character variable names

The number of five - character variable names is \(26\times36^{4}\).

Step6: Calculate six - character variable names

The number of six - character variable names is \(26\times36^{5}\).

Step7: Calculate seven - character variable names

The number of seven - character variable names is \(26\times36^{6}\).

Step8: Calculate eight - character variable names

The number of eight - character variable names is \(26\times36^{7}\).

Step9: Calculate the total number of variable names

The total number of variable names is the sum of the number of variable names of each length:
\[

$$\begin{align*} N&=26+26\times36 + 26\times36^{2}+26\times36^{3}+26\times36^{4}+26\times36^{5}+26\times36^{6}+26\times36^{7}\\ &=26\times(1 + 36+36^{2}+36^{3}+36^{4}+36^{5}+36^{6}+36^{7}) \end{align*}$$

\]
Using the sum of a geometric series formula \(S=\frac{a(r^{n}-1)}{r - 1}\) (where \(a = 1\), \(r=36\), and \(n = 8\)), we have \(1+36+36^{2}+\cdots+36^{7}=\frac{36^{8}-1}{36 - 1}=\frac{36^{8}-1}{35}\).
\[

$$\begin{align*} N&=26\times\frac{36^{8}-1}{35}\\ &=26\times\frac{2821109907456 - 1}{35}\\ &=26\times\frac{2821109907455}{35}\\ &=26\times80603140213\\ &=2095681645538 \end{align*}$$

\]

Answer:

B. 2,095,681,645,538