Exercise 9

A database uses $20$-character strings as record identifiers. The valid characters in these strings are upper-case letters in the English alphabet and decimal digits. (Recall there are $26$ letters in the English alphabet and $10$ decimal digits.) How many valid record identifiers are possible if a valid record identifier must meet all of the following criteria:

• Letter(s) from the set $\{A,E,I,O,U\}$ occur in exactly three positions of the string.

• The last three characters in the string are distinct decimal digits that do not appear elsewhere in the string.

• The remaining characters of the string may be filled with any of the remaining letters or decimal digits.