Question

a warranty identification number for a certain product consists of a letter of the alphabet followed by a six - digit number. how many possible identification numbers are there if the first digit of the six - digit number must be nonzero?
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a password is to be made from a string of five characters chosen from the lowercase letters of the alphabet and the numbers 0 through 9
(a) how many passwords are possible if there are no restrictions?
passwords
(b) how many passwords are possible if the characters must alternate between letters and numbers?
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Explanation:

Step1: Calculate letter options

There are 26 letters in the alphabet, so:
$26$

Step2: Calculate six-digit number options

First digit: 9 non-zero choices (1-9); remaining 5 digits: 10 choices each (0-9).
$9 \times 10^5 = 900000$

Step3: Total ID combinations

Multiply letter and number options:
$26 \times 900000 = 23400000$

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Step1: Total character pool size

26 lowercase letters + 10 numbers = 36 total characters.
$26 + 10 = 36$

Step2: Unrestricted password count

5 characters, no restrictions: $36^5$
$36^5 = 60466176$

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Step1: Case 1: Letter first (L-N-L-N-L)

Letters: 26 choices each; numbers:10 choices each.
$26 \times 10 \times 26 \times 10 \times 26 = 26^3 \times 10^2 = 1757600$

Step2: Case 2: Number first (N-L-N-L-N)

Numbers:10 choices each; letters:26 choices each.
$10 \times 26 \times 10 \times 26 \times 10 = 26^2 \times 10^3 = 676000$

Step3: Total alternating passwords

Add the two cases:
$1757600 + 676000 = 2433600$

Answer:

1. Warranty identification numbers: $23400000$
2. (a) Unrestricted passwords: $60466176$
3. (b) Alternating character passwords: $2433600$