We wish to make a hash table for part numbers where each part number has the format: Four lowercase vowels $($a$,$ e$,$ i$,$ o$,$ u$),$ followed by five distinct digits between $1$ and $9,$ followed by a symbol string composed of symbols from the set $\{$@$,$ $$,$ #$\}.$ The symbol string may be of any length between $2$ and $6.$

Examples:

auai$12396$@@$@##

ooee$32914$#@

eaiu$73896$$@$

ooee$32916$#@

How big should our hash table be$,$ i$.$e$.$ how many distinct part numbers are possible?

b$)$ To save storage we want to use that fact that all part numbers in actual use will have either the same number of $ and @ symbols or else one more $ than @$.$ How big should our hash table be now?