Probability Calculation A Student Chooses 5 Questions Out Of 9 True-False Questions

Hey guys, ever stumbled upon a seemingly simple math problem that makes your head spin? Well, you're not alone! Let's dive into a classic probability question that often pops up in mid-term exams. We're going to break it down step-by-step, so you'll not only understand the solution but also the logic behind it. Think of this as your friendly guide to conquering those tricky math questions. So, grab your thinking caps, and let's get started!

Understanding the Core Problem

Okay, let's break down the problem. Imagine you're a student facing a mid-term exam. This exam has 9 true-false questions, but here's the twist: you only need to pick and answer 5 of them. Sounds manageable, right? But wait, there’s a catch! The first 2 questions are compulsory, meaning you have to answer them. Now, the real question is: how many different ways can you answer this exam? This isn't just about blindly guessing true or false; it’s about figuring out the total number of possible answer combinations. This kind of problem is super common in probability and combinatorics, and it’s all about understanding the constraints and applying the right formulas. So, before we jump into calculations, let’s make sure we're crystal clear on what the problem is asking. We need to find out the total number of ways a student can answer the exam, keeping in mind those mandatory first two questions. Got it? Great, let's move on!

Breaking Down the Constraints

Let's zoom in on those constraints because they’re the key to cracking this problem. We've got 9 true-false questions staring at us, but we only need to tackle 5. Now, here's where it gets interesting: the first 2 questions are like those VIPs at a party – you have to acknowledge them. This means our choices are already partly made for us. We don't get to decide whether to answer questions 1 and 2; we just have to. So, what does this actually mean for our calculation? It means we’re not really choosing 5 questions from 9; we're essentially choosing the remaining questions from a smaller pool. Think of it like this: we've already filled 2 slots in our ‘5 questions to answer’ list. That leaves us with 3 more slots to fill. And since we've already dealt with the first 2 questions, we only have 7 questions left to choose from (the original 9 minus the 2 compulsory ones). Understanding these constraints is crucial because they directly influence how we set up our calculation. We're not dealing with a straightforward combination of 5 from 9 anymore; it’s a bit more nuanced. So, with this constraint in mind, let's see how we can actually calculate the possibilities.

Calculating Combinations: The Formula

Alright, let's get down to the nitty-gritty – the calculation part! This is where we pull out our trusty combinations formula. If you've seen this before, it might look a bit intimidating, but trust me, it's just a tool to help us count possibilities. The formula we're talking about is the combination formula, often written as nCr, where 'n' is the total number of items, and 'r' is the number of items we're choosing. In math terms, it looks like this: nCr = n! / (r! * (n-r)!). Now, what does all this mean? The "!" symbol stands for factorial, which means you multiply a number by every whole number less than it down to 1. For example, 5! is 5 x 4 x 3 x 2 x 1. So, how does this apply to our exam question? Remember, we've got 7 questions to choose from, and we need to pick 3 (since we've already committed to answering those first 2). So, our 'n' is 7, and our 'r' is 3. Plugging these numbers into our formula gives us 7C3 = 7! / (3! * 4!). Now it's just a matter of crunching those numbers. This formula is the key to figuring out how many different combinations of questions we can answer, given our constraints. Ready to see how we solve it? Let’s dive into the next step!

Step-by-Step Calculation

Okay, let's put our math hats on and break down this calculation step-by-step. We've established that we need to calculate 7C3, which, as we know, is 7! / (3! * 4!). Let’s start with the factorials. 7! (7 factorial) means 7 x 6 x 5 x 4 x 3 x 2 x 1, which equals 5040. Next, we calculate 3! (3 factorial), which is 3 x 2 x 1 = 6. And then, 4! (4 factorial) is 4 x 3 x 2 x 1 = 24. Now we have all the pieces to plug into our formula: 7C3 = 5040 / (6 * 24). Let's simplify this. 6 times 24 is 144, so we now have 7C3 = 5040 / 144. When we do the division, 5040 divided by 144, we get 35. So, 7C3 equals 35. But hold on, we're not quite at the finish line yet! This 35 tells us the number of ways we can choose the questions, but we still need to consider the true-false aspect of each question. Remember, each question has two possible answers: true or false. So, we've figured out the combinations of questions, but now we need to factor in the answer possibilities for each combination. Let's see how we do that in the next section!

Accounting for True-False Answers

Alright, we've figured out that there are 35 different ways to choose which questions to answer. Awesome! But remember, each of these questions isn't just a yes/no decision; it's a true/false situation. So, for every question we answer, there are two possibilities: we can answer it correctly with “true,” or we can answer it with “false.” This doubles the complexity, but don't worry, we've got this! Now, think about those 3 questions we're choosing from the remaining 7. Each of these questions has 2 possible answers. So, for the first question, we have 2 choices. For the second, we also have 2 choices, and for the third, yet again, 2 choices. To find the total number of ways we can answer these 3 questions, we multiply the possibilities together: 2 x 2 x 2. This gives us 2^3, which equals 8. So, for every combination of 3 questions we choose, there are 8 different ways we can answer them. This is a crucial step because it brings in the true-false element, making our calculation more accurate. Now that we know the number of question combinations and the answer possibilities for each combination, we can put it all together to find the final answer. Let’s do that now!

Calculating the Total Possibilities

Okay, we're in the home stretch now! We've done the groundwork, figured out the pieces, and now it's time to assemble the puzzle. Remember, we calculated that there are 35 different ways to choose 3 questions from the remaining 7 (after accounting for the 2 mandatory ones). And, we also figured out that for each of those 35 combinations, there are 8 different ways to answer them, considering the true-false nature of the questions. So, how do we combine these two numbers to get our final answer? Simple! We multiply them. For each of the 35 ways to choose the questions, there are 8 ways to answer them. So, the total number of possible answer combinations is 35 multiplied by 8. Let’s do that multiplication: 35 x 8. If you do the math, you'll find that 35 times 8 equals 280. But wait a second! Looking back at the original question, the answer choices don't include 280. Did we make a mistake somewhere? Not quite! We've calculated the number of ways to choose and answer the additional 3 questions. But we haven't yet accounted for the first two questions, which each have two answer possibilities (true or false). So, let’s factor those in now to get the real, final answer!

Factoring in the First Two Questions

So, we've reached a point where we've calculated 280 possibilities for answering the last three questions. Great job, guys! But let's not forget about those first two mandatory questions. They're like the foundation of our answer, and we need to make sure we've built on them correctly. Each of these first two questions, being true-false, has 2 possible answers. That means the first question can be answered in 2 ways (true or false), and the second question can also be answered in 2 ways. Now, how do we factor these possibilities into our overall calculation? Easy peasy! We simply multiply the possibilities together. We already know there are 280 ways to answer the remaining questions. For each of those ways, there are 2 ways to answer the first question, and for each of those ways, there are 2 ways to answer the second question. So, we multiply 280 by 2 (for the first question) and then by 2 again (for the second question). This looks like: 280 x 2 x 2. Let's do the math: 280 times 2 is 560, and 560 times 2 is 1120. Aha! Now we have a number that matches one of the answer choices. This final step is super important because it ensures we've considered every aspect of the problem, including those mandatory questions. So, with this in mind, let's confidently state our final answer!

The Final Answer

Alright, after all that brainpower and step-by-step calculation, we've arrived at our final answer! Remember, we started with a seemingly complex problem about a student choosing questions on an exam. We broke it down, tackled the constraints, used the combinations formula, and accounted for the true-false nature of the questions. Phew! So, what's the final verdict? We calculated that there are 1120 possible ways for the student to answer the exam, given that they have to choose 5 questions out of 9, with the first 2 being compulsory. That’s a lot of possibilities! This whole process wasn't just about getting to the right number; it was about understanding how we got there. We used a combination of logical reasoning and mathematical tools to solve the problem. And that, my friends, is the real key to conquering any math challenge. So, next time you face a similar question, remember our journey here. Break it down, consider the constraints, and take it one step at a time. You've got this! Now, let's confidently choose the correct answer from the options provided. The correct answer is (e) 1120. We nailed it!