n a symphony orchestra, there are 2 vacancies for cellists only, 4 vacancies for violinists only and 4 vacancies for any of these musicians. In how many ways can these vacancies be filled from 15 applicants, of whom 7 are violinists and 8 are cellists?

This problem can be solved using the concept of combinations in probability.

Step 1: Identify the number of ways to fill the specific vacancies for cellists and violinists.

There are 2 vacancies for cellists and we have 8 cellists. The number of ways to choose 2 cellists from 8 is given by the combination formula C(n, r) = n! / [(n-r)!r!], where n is the total number of options, r is the number of options to choose, and '!' denotes factorial.

So, C(8, 2) = 8! / [(8-2)!2!] = 28 ways.

Similarly, there are 4 vacancies for violinists and we have 7 violinists. So, C(7, 4) = 7! / [(7-4)!4!] = 35 ways.

Step 2: Identify the number of ways to fill the vacancies for any musicians.

After filling the specific vacancies, we have 9 musicians (3 cellists and 6 violinists) left and 4 vacancies for any musicians. So, C(9, 4) = 9! / [(9-4)!4!] = 126 ways.

Step 3: Multiply the results together.

The total number of ways to fill all the vacancies is the product of the number of ways to fill each set of vacancies, because these are independent events.

So, the total number of ways = 28 (for cellists) * 35 (for violinists) * 126 (for any musicians) = 123480 ways.