Question

creates a website password with six characters. there are about 3.089 × 10⁸ possible six-character passwords that use only lowercase letters. there are about 2.177 × 10⁹ possible six-character passwords that use any combination of lowercase letters, only numbers, or both lowercase letters and numbers. how many passwords can nora create that use at least one number?

Explanation:

Step1: Determine total characters available

Lowercase letters: 26, Numbers: 10. Total per character (any combination): \(26 + 10 = 36\). For 6 - character passwords with any combination: \(36^6\).

Step2: Determine passwords with only lowercase

Only lowercase letters: 26 per character. For 6 - character: \(26^6\).

Step3: Calculate passwords with at least one number

Subtract passwords with only lowercase from total any combination: \(36^6-26^6\).
Calculate \(36^6 = 36\times36\times36\times36\times36\times36=2176782336\)
Calculate \(26^6 = 26\times26\times26\times26\times26\times26 = 308915776\)
Subtract: \(2176782336 - 308915776=1867866560\approx1.87\times 10^{9}\) (Wait, but let's check the given values. Wait, the problem's given "any combination" is \(2.177\times10^{9}\) (which is \(36^6 = 2176782336\approx2.177\times10^{9}\)), "only lowercase" is \(3.089\times10^{8}\) (which is \(26^6 = 308915776\approx3.089\times10^{8}\)). So using the given approximations: \(2.177\times 10^{9}-3.089\times 10^{8}=2177000000 - 308900000 = 1868100000\approx1.87\times 10^{9}\) (but let's do it with the given numbers: \(2.177\times10^{9}-0.3089\times10^{9}=(2.177 - 0.3089)\times10^{9}=1.8681\times10^{9}\approx1.87\times 10^{9}\))

Wait, but the question is "How many passwords can Nora create that use at least one number?" So using the total (any combination: \(2.177\times10^{9}\)) minus only lowercase (\(3.089\times10^{8}\)):

First, convert \(3.089\times10^{8}\) to \(0.3089\times10^{9}\)

Then \(2.177\times10^{9}-0.3089\times10^{9}=(2.177 - 0.3089)\times10^{9}=1.8681\times10^{9}\approx1.87\times 10^{9}\) (or using exact values: \(36^6 - 26^6=2176782336 - 308915776 = 1867866560\approx1.87\times 10^{9}\))

Answer:

The number of passwords Nora can create with at least one number is approximately \(1.87\times 10^{9}\) (or exactly \(1867866560\)). If we use the given approximate values for calculation: \(2.177\times 10^{9}-3.089\times 10^{8}=1.8681\times 10^{9}\approx1.87\times 10^{9}\)