Example 2: Passwords A website requires it's users to create passwords that are 8 characters long. The password must consist of six letters and two numbers (0- 9) and none of the characters can be reused. How many different passwords can be created? Step 1: When using the fundamental counting principle with a problem with multiple components, it is helpful to make lines for each value that will be multiplied. Since this is an eight digit code, we will need eight lines. Step 2: The letters and numbers could appear in any order. We will let the first 6 values represent the letters and the last two represent the numbers. This is possible because of the commutative property of multiplication, (a . b = b - a). There are 26 letters in the alphabet, so we will enter 26 for the first value. 26 . Since we are not allow to use the first letter again, we have 25 letters left to choose. (Enter the value on the next line.) 26 Once the first two letters are used, that leaves us 24 letters for the third choice, 23 letters for the fourth choice, 22 letters of the fifth choice, and 21 letters for the sixth choice. (Enter the missing values.) 26 . 25 . 24 . . 21 . Now we need to enter the amount of numbers we have from which to choose. There are 10 numbers; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We will enter 10 in the seventh space. Since we cannot reuse the number, we will have a choice of 9 numbers left. (Enter the values below.) 26 . 25 . 24 . 23 . 22 . 21 . Step 3: Now multiply the values to get 14,918,904,000. That is almost 15 billion passwords. That is a lot of passwords! How many passwords would the website have if users were allowed to reuse the letters and numbers? (Enter the number of successful outcomes in each of the blanks below. Let the first six spaces represent letters and the last two spaces represent numbers) = 30,891,577,600 That is almost billion passwords