Twenty-three students, of whom ten are seniors, are selected as semifinalists for a literary award. Of the twenty-three students, thirteen finalists will be selected. In how many ways can the thirteen finalists contain at least four seniors?
A)1,040,754
B)474,474
C)1,106,105
D)1,098,669
E)834,834

Question

Quizplus · Accepted Answer

The total number of ways to select 13 finalists from 23 students without any restrictions is $C(23, 13)$. To find the number of ways to select at least 4 seniors among the 13 finalists, we subtract the number of ways to select 0, 1, 2, or 3 seniors from the total. This can be calculated as follows:- Total ways to select 13 finalists: $C(23, 13)$- Ways to select 0 seniors: $C(10, 0) \times C(13, 13)$- Ways to select 1 senior: $C(10, 1) \times C(13, 12)$- Ways to select 2 seniors: $C(10, 2) \times C(13, 11)$- Ways to select 3 seniors: $C(10, 3) \times C(13, 10)$The number of ways to have at least 4 seniors is the total ways minus the sum of the ways to have 0, 1, 2, or 3 seniors:$$C(23, 13) - \left( C(10, 0) \times C(13, 13) + C(10, 1) \times C(13, 12) + C(10, 2) \times C(13, 11) + C(10, 3) \times C(13, 10) \right)$$Calculating this gives the number of ways as 1,106,105.

Twenty-Three Students, of Whom Ten Are Seniors, Are Selected as Semifinalists