Question

Twenty-six students, of whom eleven are seniors, are selected as semifinalists for a literary award. Of the twenty-six students, thirteen finalists will be selected. In how many ways can the thirteen finalists contain at least five seniors?

Answer 1

Explanation:

There are total of 26 students out of which 11 are seniors

therefore remaining 15 are Juniors

There are 13 Finalist out of 26 students

We have to select 13 finalist such that there are at least five seniors

i.e. there should be minimum of 5 seniors

Possible combinations are

5 senior ,8 Junior

6 senior , 7 Junior

7 senior , 6 Junior

8 senior , 5 Junior

9 senior , 4 Junior

10 senior ,3 Junior

11 senior , 2 Junior

no of ways are
`=^(11)C_(5)* ^(15)C_(8)+^(11)C_(6)* ^(15)C_(7)+^(11)C_(7)* ^(15)C_(6)+^(11)C_(8)* ^(15)C_(5)+^(11)C_(9)* ^(15)C_(4)+^(11)C_(10)* ^(15)C_(3)+^(11)C_(11)* ^(15)C_(2)`

`=8,173,270`