In August, the department’s administrative officer, Aida Rodriguez, wrote to other department officials that a $24,666 quote for the dining room table, chairs and hutch was “a very reasonable price and the funds are available.”

“We also have a justification for the cost (as you know, the furniture hasn’t been changed since 1988) so this should not be a problem,” Ms. Rodriguez wrote to Mr. Carson’s executive assistant, Allison Mills.

Ms. Rodriguez also sent senior agency officials a detailed estimate of the dining room set provided by a Baltimore-based vendor. It included $4,097 for the 12-foot mahogany table and base, $6,488 for the “Jefferson Sideboard” and more than $7,000 for 10 chairs.

For reasons that are not clear, about $7,000 in additional costs were added when the expense was logged in a federal procurement database in December.

A department spokesman, Raffi Williams, said the money was legally allocated because the dining room table served an agencywide function and did not strictly fall under the restrictions placed on the redecoration of the secretary’s office.

Mr. Carson has said he was not aware of how expensive the dining room set was and requested that the order be canceled after the purchase, which has prompted an investigation by Republicans on the House Oversight Committee, was revealed in news reports.

The emails document a lengthy back and forth between Mr. Carson’s wife and his closest aides over the smallest details of the dining room set, including a dozen swatches detailing the fabrics available for the chairs, which were reviewed by Mrs. Carson.