A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. to do this, he ranks the orchestra's 10 cellists and 16 violinists in order of musical proficiency. what is the ratio of the total cellist rankings possible to the total violinist rankings possible?

Ranking is very critical in this problem. Therefore, the most applicable mathematical theory is permutations.

Ranking possible for the cellists:
Needed cellists = 5
Total available cellists = 10
Possible number of rankings = 10P5 = 10!/(10-5)! = 30240

Ranking possible for the violinists:
Needed violinists = 5
Total available = 16
Possible number of rankings = 16P5 = 16!/(16-5)! = 524160

Therefore,
Ratio of rankings (cellists to violinists) = 30240/524160 = 3/52