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

Answer:

123480

Step-by-step 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

$C_8^2=\dfrac{8!}{2!(8-2)!}=\dfrac{8!}{2!6!}=\dfrac{6!\cdot 7\cdot 8}{2\cdot 6!}=28$

different ways.

2. 4 violonists from 7 can be chosen in

$C_7^4=\dfrac{7!}{4!(7-4)!}=\dfrac{7!}{4!3!}=\dfrac{4!\cdot 5\cdot 6\cdot 7}{4!\cdot 2\cdot 3}=35$

different ways.

3. 4 musicians from 9 can be chosen in

$C_9^4=\dfrac{9!}{4!(9-4)!}=\dfrac{9!}{4!5!}=\dfrac{5!\cdot 6\cdot 7\cdot 8\cdot 9}{5!\cdot 2\cdot 3\cdot 4}=126$

different ways.

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