Q:

Complementary Principle

In a test of 10 questions, a student must answer exactly at least to 5 questions. How many choices do you have? R 252
What if at least 2 of the exact answers must be from the first 5 questions.
I mean, the questions are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} the right answer are: {1, 3, 7, 8, 9} or {1, 2, 3, 6, 7}.
R: 226 (hint to the second question: use the principle of complementary)
For the first part I used Com(10, 5) and it's correct. But I don't know what do do in the second part.

A:

We will look at the complimentary, that is how many possibilities you have where only zero or one are in the first five questions.
There is one way to have no answers among the first five, since it would require all the answers to be among the last five. If we place one among the first five and four among the last five, this can be done in $\binom{5}{1}\binom{5}{4}=25$ ways. So the final answer is:
$252-1-25=226$

