Question

In 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?

Answer 1

123480

Explanation:

We have to choose:

• 2 cellists from 8 cellists;
• 4 violonists from 7 violonists;
• 4 cellists or violonists from 9 remaining musicians (8-2=6 - cellists left, 7-4=3 - violonists left, 6+3=9 - left musicians in total)

1. 2 cellists from 8 can be chosen in

`C_8^2=(8!)/(2!(8-2)!)=(8!)/(2!6!)=(6!\cdot 7\cdot 8)/(2\cdot 6!)=28`

different ways.

2. 4 violonists from 7 can be chosen in

`C_7^4=(7!)/(4!(7-4)!)=(7!)/(4!3!)=(4!\cdot 5\cdot 6\cdot 7)/(4!\cdot 2\cdot 3)=35`

different ways.

3. 4 musicians from 9 can be chosen in

`C_9^4=(9!)/(4!(9-4)!)=(9!)/(4!5!)=(5!\cdot 6\cdot 7\cdot 8\cdot 9)/(5!\cdot 2\cdot 3\cdot 4)=126`

different ways.

In total, there are
`28\cdot 35\cdot 126=123480` differnt ways to fill all vacancies.