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# Permutations

## The number of arrangements when order matters

Estimated12 minsto complete
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Practice Permutations
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Estimated12 minsto complete
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Credit: Robert Lopez
Source: CK-12 Foundation

Have you ever had an account hacked? System administrators work hard to protect users from hackers that could potentially obtain your password. A few years ago, most systems would allow users to choose simple passwords consisting of only a few letters. Now, they often require you to use “at least 8 characters,” “a mix of letters and numbers,” or even “special characters.” How does this help?

#### Security in the Digital Age

One way a hacker can get into your account is by trying out all possible permutations of passwords one by one. This would take forever to do by hand, but computers can do this quickly. How long would it take a computer to discover a password by simply guessing?

Credit: Robert Lopez
Source: CK-12 Foundation

If it's a 6-letter password in which only uppercase letters can be used and repetition is allowed, there are about 309 million possible passwords. This seems like a huge number, but a computer could easily try all of those possibilities in half an hour! What happens if you increase the password size to 8 letters? There are now almost 209 billion possibilities. At the same rate as before, it would take over 14 days to crack using a computer. But computers are cheap, so it's still doable if many machines are used. Now increase the number of characters allowed to 52 (both uppercase and lowercase letters allowed); you now have about 53 trillion possibilities. Add in some numbers and special characters to the mix and it gets even stronger!

#### Explore More

1. Use permutations to find the numbers mentioned above: 309 million, 209 billion, and 53 trillion.
2. If a password is allowed to be anywhere between 8 and 10 characters (uppercase and lowercase letters and numbers allowed), how many possible passwords are there?

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