During $2023,$ Fred Reardon incurred the following expenditures in connection with his coin collecting activities:

Travel expenses $$10,000$

Purchase of coins still held at end of year $11,000$

Cost of coins sold during year $9,000$

Total $$30,000$

Fred received $$16,000$ for the coins sold during the year.

$($a$.)$ If Fred's coin collecting activities are considered to be a hobby for tax purposes, what related expenditures are deductible on his $2023$ tax return? Show computations and explain in detail.

$($b$.)$ If Fred's coin collecting activities are considered to be a trade or business for tax purposes, what related expenditures are deductible? Show computations and explain in detail.