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Chapter 10: Problem 74

Write an original problem that can be solved using the Fundamental Counting Principle. Then solve the problem.

Short Answer

Expert verified
The total number of possible passwords that can be created which are exactly 8 characters long and must include at least one of the lowercase alphabets (a-z), one of the uppercase alphabets (A-Z), and one digit from 0-9 is \( 62^8 \)

Step by step solution

01

Formulating the Problem

Imagine that we need to create a password that is exactly 8 characters long. This password must include at least one of the 26 lowercase alphabets (a-z), one of the 26 uppercase alphabets (A-Z), and one digit from 0-9. We can use any character more than once. We need to determine how many different possible passwords can be created using this criteria.
02

Applying the Fundamental Counting Principle

If we can use a character more than once, that means for each of the 8 positions, we have 62 choices (26 lowercase + 26 uppercase + 10 digits). According to The Fundamental Counting Principle the total ways is the multiplication of the number of choices for each position, so the total number of possible passwords is \(62^8 \).
03

Calculation

Calculate \(62^8 \) using a calculator.

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Most popular questions from this chapter

In Exercises \(39-44\), you are dealt one card from a 52 -card deck. Find the probability that you are dealt a 7 or a red card.

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