Question

A states's license plates consist of three letters followed by three numerals, and 243 letter arrangements are not allowed. How many plates can the state issue?

Answer 1

The number of permutations of the 26 letters of the alphabet taken 3 at a time is
\[\frac{26!}{(26-3)!}\ .........(1)\]
The number of permutations of the digits 0 to 9 taken 3 at a time is
\[\frac{10!}{(10-3)!}\ .........(2)\]
The number of permutations in (1) must be reduced by 243 to give the number of allowed permutations. Then each of the allowed permutations of letters can be taken with each of the permutations of numerals. Therefore the total number of plates that the state can issue is given by
\[(\frac{26!}{23!}-243) \times \frac{10!}{7!}=you\ can\ calculate\]