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.)

Total number of possibilities 2,095,681,645,538.

Step-by-step explanation:

The variables can be 1 to 8 characters long.

The first space can be filled by any of the 26 letters.

The remaining n places can be filled by any of the 26 letters or any of the 10 digits.

For a single character variable the number of ways to select a variable name is,

n (1 character) = 26

For two character variable the number of ways to select a variable name is,

n (2 character) = 26 × 36 = 936

For three character variable the number of ways to select a variable name is,

n (3 character) = 26 × 36 × 36 = 26 × 36² = 33,696

For four character variable the number of ways to select a variable name is,

n (4 character) = 26 × 36 × 36 × 36 = 26 × 36³ = 1,213,056

And so on.

Similarly for the eight character variable the number of ways to select a variable name is,

n (8 character) = 26 × 36 × 36... × 36 = 26 × 36⁷ = 2,037,468,266,496

Total number of possibilities 2,095,681,645,538.

d, x=8

step-by-step explanation:

simplifying

4x + -5 = 2x + 11

reorder the terms:

-5 + 4x = 2x + 11

reorder the terms:

-5 + 4x = 11 + 2x

solving

-5 + 4x = 11 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-5 + 4x + -2x = 11 + 2x + -2x

combine like terms: 4x + -2x = 2x

-5 + 2x = 11 + 2x + -2x

combine like terms: 2x + -2x = 0

-5 + 2x = 11 + 0

-5 + 2x = 11

add '5' to each side of the equation.

-5 + 5 + 2x = 11 + 5

combine like terms: -5 + 5 = 0

0 + 2x = 11 + 5

2x = 11 + 5

combine like terms: 11 + 5 = 16

2x = 16

divide each side by '2'.