This is the 4384th most frequent Russian word.
вашим
"Вашим" means "your" or "yours" (in the instrumental case, plural or formal singular).
'Вашим' is used here as the instrumental case of the possessive pronoun 'ваш', modifying 'другом' to indicate 'by your friend'.
Вашим другом оказался мой брат.
Your friend turned out to be my brother.
'Вашим' functions as the instrumental case of 'ваш', modifying 'разрешением' to express 'with your permission'.
С вашим разрешением, мы начнем собрание.
With your permission, we will start the meeting.
'Вашим' serves as the instrumental case here with 'участием', showing the means or tool—in this instance, 'by your participation'.