This week's problem is in honor of Elvis Presley, whose birthday is this week (JANUARY 8th). According to experts in computer security, you should use at least an eight-letter password, using a combination of upper-case letters, lower-case letters, numbers, and some punctuation symbols. How many eight-letter passwords are possible that contain the name ELVIS? The only restriction is that ELVIS must appear in the password as five consecutive letters (in order) but the letters may be any combination of upper-case and lower-case. Symbols may repeat. You may choose from 72 different symbols on the keyboard (26 upper-case letters, 26 lower-case letters, 10 digits, and 10 punctuation symbols).

Solution to the Problem: The answer is 47,775,744 different passwords! The letters for ELVIS can fit in the eight-letter password in four different ways: E L V I S _ _ _ _ E L V I S _ _ _ _ E L V I S _ _ _ _ E L V I S To figure out the number of ways that ELVIS could appear at the beginning of your password, there are 2 ways that E could be chosen (upper and lower case), 2 ways that L could be chosen, 2 ways that V could be chosen, 2 ways that I could be chosen, 2 ways that S could be chosen, and then 72 ways that the last three symbols could be chosen, for a total of 11,943,936 permutations. 2 2 2 2 2 72 72 72 = 11,943,936 The other three positions of ELVIS result in the same number of passwords, so the total number is 11,943,936 + 11,943,936 + 11,943,936 +11,943,936 = 47,775,744 different passwords!

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