An exam consists of six true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways may the exam be completed if a penalty is imposed for each incorrect answer, so that a student may leave some questions unanswered?

Since there are 2 choices for each answer, there are 2^6 = 64 ways to answer all the questions.

If some may be left unanswered, then we have

2^6 * 6C6 ways to select and answer all 6
2^5 * 6C5 ways to select and answer 5
2^4 * 6C4 ways to select and answer 4
and so on. Adding them all up, you get 728

Well, let me juggle this question for you!

In the first case, if every question must be answered, there are 2 answer choices for each question, true or false. So, for each question, there are 2 possible ways to answer it. Since there are 6 questions in total, the number of different ways a student can complete the exam is 2^6, which is 64.

Now, in the second case, where a penalty is imposed for each incorrect answer and students can leave some questions unanswered, it gets a bit trickier. Let's think about it this way: for each question, a student has 3 choices - they can answer it correctly, answer it incorrectly (and incur a penalty), or leave it unanswered.

So, for 6 questions, each having 3 choices, the total number of ways the exam can be completed is 3^6, which is 729.

So, the answer is 64 ways without a penalty, and 729 ways with a penalty. Now, go ace that exam like a juggling superstar!

To calculate the number of different ways a student can complete the exam, we need to consider that each question has two options: true or false. Since there are six questions in total, each with two options, the total number of different ways to complete the exam is 2^6, which is equal to 64.

However, if a penalty is imposed for each incorrect answer and a student is allowed to leave some questions unanswered, then the number of ways to complete the exam changes.

Let's analyze this scenario:

For each question, a student has three options: true, false, or not answering the question. Therefore, for each question, there are three possibilities.

Since there are six questions in total, the total number of ways to complete the exam with the penalty and the option to leave questions unanswered is 3^6, which is equal to 729.

To find out the number of different ways a student can complete the exam, we need to consider the two possibilities for each question: either it is answered correctly or incorrectly. Since there are six questions, and each question has two possibilities, the total number of different ways to complete the exam can be calculated by multiplying the number of possibilities for each question:

Number of ways = 2 * 2 * 2 * 2 * 2 * 2 = 2^6 = 64.

Therefore, there are 64 different ways a student can complete the exam if they must answer every question.

Now, let's consider the second scenario where a penalty is imposed for each incorrect answer, allowing students to leave some questions unanswered.

For each question, there are three possibilities: it can be answered correctly, answered incorrectly, or left unanswered. Since there are six questions, the total number of ways to complete the exam in this scenario can be calculated by multiplying the number of possibilities for each question:

Number of ways = 3 * 3 * 3 * 3 * 3 * 3 = 3^6 = 729.

Therefore, there are 729 different ways the exam can be completed if students can leave some questions unanswered.