Question

nora creates a website password with six characters. there are about (3.089 \times 10^8) possible six-character passwords that use only lowercase letters. there are about (2.177 \times 10^9) possible six-character passwords that use any combination of only lowercase letters, only numbers, or both lowercase letters and numbers. how many passwords can nora create that use at least one number?

Explanation:

Step1: Understand the problem

We need to find the number of six - character passwords that use at least one number. The total number of six - character passwords that use any combination of lowercase letters and numbers is given as \(2.177\times10^{9}\), and the number of six - character passwords that use only lowercase letters is \(3.089\times10^{8}\).

Step2: Use the principle of complementary counting

The number of passwords with at least one number is equal to the total number of passwords (using letters and numbers) minus the number of passwords with only letters.

First, rewrite the numbers in standard form or make the exponents the same for subtraction. Let's convert \(3.089\times 10^{8}\) to \(0.3089\times10^{9}\).

Then, subtract: \((2.177\times 10^{9})-(0.3089\times 10^{9})=(2.177 - 0.3089)\times10^{9}\)

Calculate \(2.177-0.3089 = 1.8681\)

So the number of passwords with at least one number is \(1.8681\times 10^{9}\)

Answer:

\(1.8681\times 10^{9}\)