You need to pay attention to three things:
1. Do you have vacation days left?
Your supervisor decides whether you may take vacation days and checks if you have enough left. After your application for business travel has been signed by your supervisor, the number of vacation days is only relevant for the calculation of your reimbursement.
2. Price comparison
Whenever you combine private vacation with business travel, to calculate the reimbursement first a hypothetical travel itenerary is established (how would the trip have been carried out if no vacation were involved). Only the costs of the hypothecical trip can be reimbursed, unless the actual costs were even lower.
Therefore, it is mandatory to carry out a price comparison on the days the actual tickets were booked and attach it to the reimbursement application (e.g Screenshots, quote from a travel agency, etc.).
Example: Travel to X from Berlin - Conference at X - Travel to Y - vacation in Y - travel from Y to Berliin
The hypothetical itenary would have shown a travel back to Berlin directly after the end of the conference from place X. The price comparison needs to be carried out for that day and that route. Same if the vacation had taken place also at X.
3. Limit of 5 working days
If (days actually away) minus (days hypothetically away) is greater than 5 working days, it is assumed that the trip was mainly in private, not in business, interest. In that case, the costs of travelling to and from the location will not be reimbursed. Especially for long routes (e.g. flight overseas), this can mean significant financial disadvantages.
Attention: If you are in any way unsure whether your vacation is longer than 5 days, please contact the travel department directly and ask them for written approval of your travel plans.
More information can be found in this memo from the travel department: hier (official German) and here (unofficial translation).