Official Explanation
The requirement to select 4 books, including 2 or more biographies, means that you have to consider three cases. A student can choose 4 biographies and no novels, or 3 biographies and 1 novel, or 2 biographies and 2 novels.
Case 1: Choose 4 biographies. This case is easy, as there is only 1 way to choose all four biographies and no novels.
In the other two cases, you have to find the number of ways of choosing the biographies and the number of ways of choosing the novels and then multiply these two numbers.
Case 2: Choose 3 biographies and 1 novel. First, you need to find the number of ways of choosing 3 biographies out of 4. If you think of this as not choosing 1 out of the 4, you see that there are 4 choices. The number of ways of choosing 1 novel out of the 6 novels is 6. Therefore, the total number of choices is (4)(6) = 24.
Case 3: Choose 2 biographies and 2 novels. First, you need to find the number of ways of choosing 2 biographies out of 4. This number is sometimes called “4 choose 2” or the number of combinations of 4 objects taken 2 at a time. If you remember the combinations formula, you know that the number of combinations is \(\frac{4!}{2!(4-2)!}\) (which is denoted symbolically as \(4C_{2}\)). The value of \(\frac{4!}{2!(4-2)!}\) is \(\frac{4*3*2!}{2*2!} = \frac{4*3}{2} = 6\). Thus, there are 6 ways to choose 2 biographies out of 4. Similarly, the number of ways to choose 2 novels out of 6 is \(\frac{6!}{2!4!} = \frac{6*5}{2} = 15\) . Thus, the total number of ways to choose 2 biographies and 2 novels is \((6)(15) = 90.\)
Adding the number of ways to choose the books for each of the three cases, you get a total of \(1 + 24 + 90 = 115.\)
The correct answer is Choice B.