A test contains 10 T/F questions, 5 must be marked true, and 5 false...

Question

A quiz consists of ten true/false questions.

a) In how many distinct ways can the quiz be completed if no answers are left blank?

b) In how many ways can the quiz be completed if five questions must be marked true and the other five must be marked false?

I know the answer to (a) is 1024 and (b) is 252.

I got (a) by calculating $2^{10} = 1024$.

But for (b) I am not sure how get the correct answer. Could someone please walk me through this?

Answer 1

For part b), what you are really saying is that you choose any $5$ questions to be true, and you can do this in ${10 \choose 5 }= 252$ ways. But once you have chosen $5$ true questions, you know that the remaining answers have to be false, so the answer is $252$.

Answer 2

(b) Is equivalent to chose $5$ answers (to be marked true) out of $10$. The rest are to be marked false, thus we obtain $$\binom{10}5 = \frac{9\cdot8\cdot7\cdot6\cdot5}{5\cdot4\cdot3\cdot2\cdot1} = 252$$

Answer 3

Hint:

Of the "$10$" answers, you need to "choose $5$" of them to be true.