A reading list for humanities course consists of 10 books, of which 4 are biographies and the rest are novels. Each student is required to read a selection of 4 books from the list, including 2 or more biographies. How many selections of 4 books satisfy the requirements?
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To find the number of selections of 4 books that satisfy the requirements, we need to consider two scenarios:
1. Selecting 2 biographies and 2 novels.
2. Selecting all 4 biographies.
Scenario 1: Selecting 2 biographies and 2 novels:
There are 4 biographies to choose from, and we need to select 2 of them. This can be done in C(4,2) = 6 ways.
There are (10 - 4) = 6 novels to choose from, and we need to select 2 of them. This can be done in C(6,2) = 15 ways.
The total number of selections of 2 biographies and 2 novels is 6 * 15 = 90.
Scenario 2: Selecting all 4 biographies:
There are 4 biographies to choose from, and we need to select all of them. This can be done in C(4,4) = 1 way.
To find the total number of selections that satisfy the requirements, we need to consider both scenarios and add the results:
Total = 90 + 1 = 91.
Therefore, there are 91 selections of 4 books that satisfy the requirements.
To find the number of selections of 4 books that satisfy the requirements, we need to consider two cases:
Case 1: Both biographies are chosen.
In this case, we need to select 2 more books from the remaining novels. There are (4 choose 2) = 6 ways to choose 2 novels from the remaining 6 novels.
Case 2: One biography is chosen.
In this case, we need to select 1 more biography and 2 more books from the remaining novels. There are (4 choose 1) = 4 ways to choose 1 more biography from the remaining 3 biographies, and (6 choose 2) = 15 ways to choose 2 novels from the remaining 6 novels.
Therefore, the total number of selections of 4 books that satisfy the requirements is 6 + 4 * 15 = 66.
So, there are 66 selections of 4 books from the reading list that satisfy the requirements.
There are 4C2 = 6 ways to choose the two required biographies.
That leaves 8 other books, with 8C2 = 28 ways to pick them.
So, there are 6*28 = 168 ways to pick 4 books as required.