For all problems: SHOW YOUR WORK! SHOW YOUR WORK! Answers without shown work are worth no points.

1. (20 points) A Vehicle Identification Number (VIN) is a unique code (e.g.: 1G6AF5SX6D0125409) used by the automotive industry to identify individual vehicles. Since 1981, each on-road vehicle built and sold in the United States typically (exceptional cases disregarded for the purpose of this exercise) is required to contain a 17-character VIN structured and constrained as follows:

~ A three-character (alpha-numeric) world manufacturer identifier (WMI). Excludes I, O & Q

~ A five-character (alpha-numeric) vehicle descriptor (VD). Excludes I, O & Q

~ A single (numeric or 'X') check digit (CD). The digit is calculated from the other 16 letters/digits (and is used for detection of invalid VIN).

~ A one-character (alpha-numeric) model year (MY). Excludes 'I', 'O', 'Q', 'U', 'Z', & '0' (zero)

~ A one-character (alpha-numeric) plant code (PC). Excludes I, O & Q

~ A six-character (alpha-numeric) serial number (SN). Excludes I, O & Q

Auxiliary notes:

~ only UPPERCASE letters allowed

~ The usual 10 decimal numeral digits (0-9) are allowed except for model year, for which '0' is not allowed.

~ None of the letters I, O, or Q are allowed for any part of the VIN

~ 'U' and 'Z' (in addition to the 'I', 'O', 'Q' and '0' noted above) are also not allowed for model year

~ Serial number is alpha-numeric in general (i.e., not necessarily numeric although it is most commonly so).

From the above, compute the theoretical maximum number of (distinct) VINs the underlying structure would be able to support.

HINT: Since the check digit is calculated from the other 16 letters/digits, would it be a factor in determining the maxim

NOTE: You can leave your answer as an expression involving terms containing numbers raised to certain powers.