💡Okay, so I have this problem here: Given that x times (1 + i) equals 1 + y times i, where x and y are real numbers. I need to find the value of the magnitude of x + y times i. Hmm, okay, let me try to figure this out step by step.First, I remember that when dealing with complex numbers, you can separate the real and imaginary parts. So, if I have an equation involving complex numbers, the real parts must be equal, and the imaginary parts must be equal. That seems like a good starting point.So, let me write down the given equation again: x(1 + i) = 1 + y i. I need to expand the left side to see what the real and imaginary parts are. If I distribute x over (1 + i), I get x times 1 plus x times i, which is x + x i. Okay, so the left side is x + x i.Now, the right side of the equation is 1 + y i. So, comparing the real parts and the imaginary parts separately, I can set up two equations. The real part on the left is x, and on the right, it's 1. So, x must equal 1. That seems straightforward.Next, the imaginary part on the left side is x, and on the right side, it's y. So, x must equal y as well. Since I already found that x is 1, that means y is also 1. Okay, so x is 1 and y is 1. That seems pretty clear.Now, the problem asks for the magnitude of x + y i. The magnitude of a complex number a + b i is given by the square root of (a squared plus b squared). So, in this case, a is x, which is 1, and b is y, which is also 1. Therefore, the magnitude should be the square root of (1 squared plus 1 squared).Calculating that, 1 squared is 1, and 1 squared is also 1. Adding those together gives 2. Taking the square root of 2 gives me the magnitude. So, the magnitude of x + y i is the square root of 2.Wait, let me just double-check to make sure I didn't make any mistakes. Starting from the beginning, x(1 + i) equals 1 + y i. Expanding the left side gives x + x i. Equating real parts, x equals 1. Equating imaginary parts, x equals y, so y is also 1. Then, the magnitude is sqrt(1^2 + 1^2) which is sqrt(2). Yeah, that seems right.I guess another way to think about it is by using complex number properties. If I have x(1 + i) = 1 + y i, I can solve for x by dividing both sides by (1 + i). But dividing by a complex number might be a bit more involved, but let's try it.So, x equals (1 + y i) divided by (1 + i). To simplify this, I can multiply the numerator and the denominator by the conjugate of the denominator, which is (1 - i). That way, I can eliminate the imaginary unit from the denominator.Multiplying out the numerator: (1 + y i)(1 - i) equals 1 times 1 plus 1 times (-i) plus y i times 1 plus y i times (-i). That simplifies to 1 - i + y i - y i squared. Since i squared is -1, that becomes 1 - i + y i + y. Combining like terms, the real parts are 1 + y, and the imaginary parts are (-1 + y)i.The denominator is (1 + i)(1 - i), which is 1 squared minus i squared, which is 1 - (-1) = 2.So, x equals (1 + y + (y - 1)i) divided by 2. Since x is a real number, the imaginary part must be zero. Therefore, y - 1 must equal zero, which means y equals 1. Plugging that back into the real part, 1 + y equals 2, so x equals 2 divided by 2, which is 1. So, again, x is 1 and y is 1. That confirms my earlier result.Therefore, the magnitude of x + y i is sqrt(1^2 + 1^2) = sqrt(2). I think that's solid. I don't see any mistakes in my reasoning, and both methods lead me to the same conclusion. So, I'm pretty confident that the magnitude is sqrt(2).Just to recap, the key steps were:1. Expand the left side of the equation to separate real and imaginary parts.2. Set the real parts equal and the imaginary parts equal to solve for x and y.3. Use the formula for the magnitude of a complex number to find |x + y i|.Everything checks out, so I think I'm done here.

💡Alright, so I have this problem about estimating the probability of getting an excellent result in a dart-throwing round. Let me try to understand what's being asked here.First, the problem defines a "round" as throwing the dart three times. For it to be considered "excellent," at least two out of those three throws need to land in the 8th ring or higher. Okay, so that means if two or all three throws hit the 8th ring or higher, it's excellent.To estimate this probability, they're using a random simulation experiment. The computer generates random numbers, either 0 or 1, based on the player's previous performance. Here, 0 means the throw didn't land in the 8th ring or higher, and 1 means it did. So, each 0 or 1 represents the outcome of a single throw.Then, they group these random numbers into sets of three, each group representing the result of one round. So, each group of three 0s and 1s corresponds to three throws in a round.After running the simulation, they got 20 such groups. The groups are listed as:101, 111, 011, 101, 010, 100, 100, 011, 111, 110,000, 011, 010, 001, 111, 011, 100, 000, 101, 101So, there are 20 rounds simulated here. My task is to estimate the probability that a single round is excellent, meaning at least two of the three throws are 1s (i.e., landed in the 8th ring or higher).Alright, so I need to go through each of these 20 groups and count how many of them have at least two 1s. Then, the probability would be the number of excellent rounds divided by the total number of rounds, which is 20.Let me list out all the groups and count the number of 1s in each:1. 101: Let's see, the digits are 1, 0, 1. So, that's two 1s.2. 111: All three are 1s. That's three 1s.3. 011: The digits are 0, 1, 1. That's two 1s.4. 101: Again, two 1s.5. 010: The digits are 0, 1, 0. That's one 1.6. 100: The digits are 1, 0, 0. That's one 1.7. 100: Same as above, one 1.8. 011: Two 1s.9. 111: Three 1s.10. 110: Two 1s.11. 000: No 1s.12. 011: Two 1s.13. 010: One 1.14. 001: One 1.15. 111: Three 1s.16. 011: Two 1s.17. 100: One 1.18. 000: No 1s.19. 101: Two 1s.20. 101: Two 1s.Now, let's count how many of these have at least two 1s. Going through the list:1. 101: Yes (2)2. 111: Yes (3)3. 011: Yes (2)4. 101: Yes (2)5. 010: No (1)6. 100: No (1)7. 100: No (1)8. 011: Yes (2)9. 111: Yes (3)10. 110: Yes (2)11. 000: No (0)12. 011: Yes (2)13. 010: No (1)14. 001: No (1)15. 111: Yes (3)16. 011: Yes (2)17. 100: No (1)18. 000: No (0)19. 101: Yes (2)20. 101: Yes (2)Now, let's count the number of "Yes" responses:1. Yes2. Yes3. Yes4. Yes5. No6. No7. No8. Yes9. Yes10. Yes11. No12. Yes13. No14. No15. Yes16. Yes17. No18. No19. Yes20. YesSo, let's count them:1. Yes (1)2. Yes (2)3. Yes (3)4. Yes (4)8. Yes (5)9. Yes (6)10. Yes (7)12. Yes (8)15. Yes (9)16. Yes (10)19. Yes (11)20. Yes (12)So, there are 12 rounds out of 20 that are considered excellent.Therefore, the estimated probability is the number of excellent rounds divided by the total number of rounds, which is 12/20.Simplifying that, 12 divided by 20 is 0.6.So, the estimated probability is 0.6.I think that's it. I went through each group, counted the number of 1s, determined if it was excellent, and then calculated the probability based on the simulation results.

💡Okay, so I have this identity to prove: the sum from k equals 0 to n of C(n, k) times C(m + k, r) times (-1)^k equals (-1)^n times C(m, r - n). Hmm, that's a bit intimidating at first glance, but let's break it down step by step.First, I need to recall what these binomial coefficients represent. C(n, k) is the number of ways to choose k elements from a set of n elements. Similarly, C(m + k, r) is the number of ways to choose r elements from a set of m + k elements. The (-1)^k term suggests that this might be an alternating sum, which often comes up in inclusion-exclusion principles or generating functions.I wonder if generating functions could be useful here. I remember that generating functions can turn sums involving binomial coefficients into products, which might simplify things. Let me think about how to set that up.If I consider the generating function for C(n, k), that's just (1 + x)^n. Similarly, the generating function for C(m + k, r) might be a bit trickier. Wait, C(m + k, r) is the same as the coefficient of x^r in (1 + x)^{m + k}. So, if I have a sum over k of C(n, k) times C(m + k, r) times (-1)^k, maybe I can express this as a coefficient extraction.Let me write that out. The sum is equal to the sum from k=0 to n of C(n, k) * [x^r] (1 + x)^{m + k} * (-1)^k. Since [x^r] is a linear operator, I can interchange the sum and the coefficient extraction. So, this becomes [x^r] sum from k=0 to n of C(n, k) * (1 + x)^{m + k} * (-1)^k.Now, let's look at the sum inside. That's sum from k=0 to n of C(n, k) * (1 + x)^{m + k} * (-1)^k. I can factor out (1 + x)^m, so it becomes (1 + x)^m * sum from k=0 to n of C(n, k) * (1 + x)^k * (-1)^k.Wait a minute, the sum from k=0 to n of C(n, k) * (1 + x)^k * (-1)^k looks like the binomial expansion of (1 - (1 + x))^n. Because (a + b)^n is sum from k=0 to n of C(n, k) a^{n - k} b^k. If I set a = 1 and b = -(1 + x), then it becomes (1 - (1 + x))^n, which simplifies to (-x)^n.So, putting it all together, the sum becomes (1 + x)^m * (-x)^n. Therefore, the entire expression is [x^r] (1 + x)^m * (-x)^n. Let's write that as [x^r] (-1)^n x^n (1 + x)^m.Now, I need to find the coefficient of x^r in (-1)^n x^n (1 + x)^m. That's the same as (-1)^n times the coefficient of x^{r - n} in (1 + x)^m. The coefficient of x^{r - n} in (1 + x)^m is just C(m, r - n), provided that r - n is between 0 and m. Since the problem states that m >= r >= n, r - n is non-negative and less than or equal to m, so that's valid.Therefore, the coefficient is (-1)^n * C(m, r - n), which is exactly what we wanted to prove. So, the original sum equals (-1)^n * C(m, r - n).Wait, let me double-check my steps to make sure I didn't make a mistake. I used generating functions, interchanged the sum and coefficient extraction, recognized the binomial expansion, simplified, and then extracted the coefficient. Each step seems logical, and I don't see any errors in the reasoning. Yeah, I think this works.

💡First, I need to determine the total amount of material Cheryl bought. She purchased 5/11 square yards of one type of material and 2/3 square yards of another type. To add these fractions, I'll find a common denominator, which is 33.Converting 5/11 to a fraction with a denominator of 33:5/11 = 15/33Converting 2/3 to a fraction with a denominator of 33:2/3 = 22/33Adding these together:15/33 + 22/33 = 37/33 square yardsCheryl used 0.6666666666666665 square yards of material, which is equivalent to 2/3 or 22/33 square yards.Subtracting the used material from the total purchased:37/33 - 22/33 = 15/33 square yardsSimplifying 15/33 gives 5/11. Therefore, Cheryl has 5/11 square yards of material left.

💡First, I need to determine the number of bike riders at Minewaska State Park. I'll start by defining variables for the number of bike riders and hikers. Let ( B ) represent the number of bike riders and ( H ) represent the number of hikers.From the problem, I know two key pieces of information:1. There are 178 more hikers than bike riders, which can be expressed as ( H = B + 178 ).2. The total number of people at the park is 676, so ( H + B = 676 ).Next, I'll substitute the expression for ( H ) from the first equation into the second equation:[(B + 178) + B = 676]Combining like terms gives:[2B + 178 = 676]To isolate ( B ), I'll subtract 178 from both sides of the equation:[2B = 498]Finally, I'll divide both sides by 2 to find the value of ( B ):[B = 249]Therefore, there were 249 people riding their bikes at Minewaska State Park that day.

💡Okay, so I have this question about correlation coefficients, which is something I remember from my statistics class. Let me try to work through it step by step.First, let me read the question again carefully. It says:Two variables x, y and their correlation coefficient r have the following statements:1. If r > 0, then as x increases, y also increases correspondingly;2. If r < 0, then as x increases, y also increases correspondingly;3. If r = 1 or r = -1, then the relationship between x and y is completely corresponding (there is a functional relationship), and all the scatter points on the scatter plot are on a straight line.Among these, the correct ones are (　　)A: ①② B: ②③ C: ①③ D: ①②③So, I need to determine which of these statements are correct. Let me recall what I know about correlation coefficients.The correlation coefficient, r, measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. - If r is positive, it means that as one variable increases, the other tends to increase as well. This is called a positive correlation.- If r is negative, it means that as one variable increases, the other tends to decrease. This is called a negative correlation.- If r is exactly 1 or -1, it means there is a perfect linear relationship between the variables. All the data points lie exactly on a straight line.Now, let's analyze each statement one by one.**Statement 1: If r > 0, then as x increases, y also increases correspondingly.**Hmm, this sounds familiar. If the correlation coefficient is positive, does that mean y increases as x increases? Yes, that's what positive correlation means. So, I think this statement is correct.**Statement 2: If r < 0, then as x increases, y also increases correspondingly.**Wait, this seems off. If r is negative, that should mean that as x increases, y decreases, right? Because negative correlation implies an inverse relationship. So, if x goes up, y goes down. Therefore, this statement is incorrect because it says y increases when x increases, which is the opposite of what a negative correlation indicates.**Statement 3: If r = 1 or r = -1, then the relationship between x and y is completely corresponding (there is a functional relationship), and all the scatter points on the scatter plot are on a straight line.**Okay, so if r is exactly 1 or -1, does that mean all the points lie on a straight line? Yes, that's correct. A perfect correlation, whether positive or negative, means there's a linear relationship without any scatter. So, this statement is correct.So, summarizing:- Statement 1: Correct- Statement 2: Incorrect- Statement 3: CorrectTherefore, the correct statements are ① and ③, which corresponds to option C.Wait, let me double-check to make sure I didn't make a mistake. For statement 2, if r is negative, it's a negative correlation, so as x increases, y should decrease. So, the statement says y increases, which is wrong. So, statement 2 is definitely incorrect.And for statement 3, perfect correlation does mean all points lie on a straight line, so that's correct.Yeah, I think I'm confident now. So, the correct answer should be option C: ①③.