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Counting the number of ways a test can be answered is at least one T/F and at least one multiple choice question must be answered
An exam has $15$ questions: eight true/false questions and seven multiple choice questions. You are asked to answer five of them, but your professor requires you to answer at least one true/false question and at least one multiple choice question. How many ways can you choose the questions you plan to answer.
step1. I chose $1$ question of true false type. $8$ ways of doing so.
step2. I chose $1$ question of MCQ type. $7$ ways of doing so.
step3. I chose $3$ questions of the remaining $13$ questions. $13 \cdot 12 \cdot 11$ ways of doing so.
step4. Hence, $8 \cdot 7 \cdot 13 \cdot 12 \cdot 11$ gives number of sequences of $5$ questions from a collection of $15$ questions such that there is at least one question from t/f type and mcq type each.
step5. Let the number of collections of $5$ questions from a collection of $15$ questions such that there is at least one question from t/f type and mcq type each be $N$.
step6. $N \cdot 5! = 8 \cdot 7 \cdot 11 \cdot 12 \cdot
You overcount all of the cases. For a case where you answer two true/false questions and three multiple choice questions, there are two way to pick which one you count in step 1 and three in step 2, so this will get counted six times.
The easier way is to count the number of ways you can choose any five questions, then subtract the ones that are all the same kind.2018-05-16 15:20:02