answer：To determine the number of possible monograms for the baby Zeta, we need to consider the following:1. **Fix the Last Initial**: The last initial is fixed as 'Z' because the baby's last name is Zeta.2. **Choose Initials for First and Middle Names**: We need to choose two distinct letters from the remaining 25 letters of the alphabet (excluding 'Z') for the first and middle initials. These two letters must be in alphabetical order.3. **Counting the Combinations**: The number of ways to choose 2 letters out of 25, where order does not matter (since the order is automatically alphabetical), is given by the combination formula: [ binom{n}{k} = frac{n!}{k!(n-k)!} ] where ( n ) is the total number of items to choose from, and ( k ) is the number of items to choose. Here, ( n = 25 ) and ( k = 2 ): [ binom{25}{2} = frac{25 times 24}{2 times 1} = 300 ]4. **Conclusion**: There are 300 ways to choose two distinct letters from the 25 letters (excluding 'Z') to form the first and middle initials in alphabetical order. Therefore, there are 300 possible monograms for the baby Zeta.Thus, the number of such monograms possible is (boxed{text{(B) } 300}).

answer：First, I recognize that the H.C.F. of the three numbers is 210, meaning each number is a multiple of 210. The L.C.M. of the numbers is provided to have additional factors of 11, 17, and 23. Since these are prime numbers and do not divide into 210, they are unique to the L.C.M.To find the L.C.M., I multiply the H.C.F. by these additional factors:[text{L.C.M.} = 210 times 11 times 17 times 23]Calculating this step-by-step:[210 times 11 = 2310][2310 times 17 = 39270][39270 times 23 = 902910]So, the L.C.M. is 902910.Since the L.C.M. is a multiple of the largest number, the largest number must be the product of the H.C.F. and the largest unique factor of the L.C.M., which is 23:[text{Largest number} = 210 times 23 = 4830]

answer：First, I need to determine how many positive integers less than 500 are multiples of 7. To do this, I divide 500 by 7 and take the floor of the result to find the largest integer multiple of 7 that is less than 500.Next, I need to exclude the numbers that are multiples of both 7 and 14, which are essentially the multiples of 14. I do this by dividing 500 by 14 and taking the floor of the result to find the largest integer multiple of 14 that is less than 500.Finally, I subtract the number of multiples of 14 from the number of multiples of 7 to get the count of numbers that are multiples of 7 but not multiples of 14.

answer：Okay, so I have this problem here where P varies directly as Q and inversely as U. Hmm, I remember from class that when something varies directly, it means if one goes up, the other goes up too, right? And inversely means if one goes up, the other goes down. So, putting that together, P is directly proportional to Q and inversely proportional to U. The problem gives me that when P is 12 and U is 16/25, Q is 1/2. Then it asks me to find Q when P is 27 and U is 9/49. Alright, let's break this down step by step.First, I think I need to write the relationship as an equation. Since P varies directly as Q and inversely as U, that should be something like P equals some constant k times Q divided by U. So, P = k * (Q/U). Yeah, that makes sense because if Q increases, P increases, and if U increases, P decreases, which fits the direct and inverse variations.Now, I need to find the constant k. They gave me a specific case where P is 12, U is 16/25, and Q is 1/2. So, plugging those values into the equation: 12 = k * (1/2) / (16/25). Let me compute that.First, dividing 1/2 by 16/25 is the same as multiplying 1/2 by 25/16. So, (1/2) * (25/16) = 25/32. So, now the equation is 12 = k * (25/32). To solve for k, I can multiply both sides by 32/25. So, k = 12 * (32/25). Let me calculate that.12 times 32 is 384, and 384 divided by 25 is 15.36. Hmm, 15.36 is a decimal, but maybe I should keep it as a fraction. 12 is 12/1, so 12 * 32 is 384, and 384 divided by 25 is 384/25. So, k is 384/25. Okay, that's the constant.Now, the problem asks for Q when P is 27 and U is 9/49. So, I need to use the same relationship: P = k * (Q/U). I have P as 27, k as 384/25, and U as 9/49. I need to solve for Q.So, plugging in the values: 27 = (384/25) * (Q / (9/49)). Let me simplify this step by step. Dividing by 9/49 is the same as multiplying by 49/9. So, Q divided by 9/49 is Q * (49/9). So, the equation becomes 27 = (384/25) * (49/9) * Q.Let me compute (384/25) * (49/9). First, multiply the numerators: 384 * 49. Hmm, 384 times 49. Let me compute that. 384 * 50 would be 19,200, so subtract 384 to get 19,200 - 384 = 18,816. So, the numerator is 18,816.Now, the denominator is 25 * 9, which is 225. So, the fraction is 18,816/225. So, now the equation is 27 = (18,816/225) * Q.To solve for Q, I need to divide both sides by (18,816/225). Dividing by a fraction is the same as multiplying by its reciprocal. So, Q = 27 * (225/18,816).Let me compute that. 27 times 225 is... 27 * 200 is 5,400, and 27 * 25 is 675, so total is 5,400 + 675 = 6,075. So, the numerator is 6,075, and the denominator is 18,816. So, Q = 6,075 / 18,816.Now, I need to simplify this fraction. Let's see if 6,075 and 18,816 have a common factor. Let's try dividing both by 3. 6,075 divided by 3 is 2,025, and 18,816 divided by 3 is 6,272. So, now it's 2,025 / 6,272.Let me check if these can be simplified further. 2,025 divided by 3 is 675, and 6,272 divided by 3 is 2,090.666..., which is not a whole number, so 3 doesn't work anymore. Let's try 5. 2,025 ends with a 5, so divided by 5 is 405. 6,272 divided by 5 is 1,254.4, which isn't a whole number. So, 5 doesn't work.How about 9? 2,025 divided by 9 is 225, and 6,272 divided by 9 is approximately 696.888..., which isn't a whole number. So, 9 doesn't work. Maybe 15? 2,025 divided by 15 is 135, and 6,272 divided by 15 is about 418.133..., not a whole number.Wait, maybe I made a mistake earlier. Let me double-check my calculations. When I had 6,075 / 18,816, I divided both by 3 to get 2,025 / 6,272. Maybe I should try dividing both by a larger number. Let's see, 2,025 and 6,272. Let me see if they have a common factor of 3 again. 2,025 is divisible by 3, but 6,272 divided by 3 is 2,090.666..., which isn't a whole number. So, no.Wait, maybe I can factor both numbers. Let's factor 2,025. 2,025 divided by 5 is 405, divided by 5 again is 81, which is 9 squared. So, 2,025 is 5^2 * 9^2. Now, 6,272. Let's see, 6,272 divided by 2 is 3,136, divided by 2 is 1,568, divided by 2 is 784, divided by 2 is 392, divided by 2 is 196, divided by 2 is 98, divided by 2 is 49, which is 7 squared. So, 6,272 is 2^7 * 7^2.So, the prime factors of 2,025 are 5^2 * 3^4, and the prime factors of 6,272 are 2^7 * 7^2. There are no common factors between them, so the fraction 2,025 / 6,272 is already in its simplest form.Wait, but looking back at the answer choices, I see options like 225/696, which is one of the choices. Hmm, 225 is 2,025 divided by 9, and 696 is 6,272 divided by 9. But earlier, when I tried dividing by 9, I got 225 / 696.888..., which wasn't a whole number. Wait, maybe I made a mistake in my division.Let me check 6,272 divided by 9. 9 times 696 is 6,264, and 6,272 minus 6,264 is 8, so it's 696 and 8/9. So, it's not a whole number. So, 225/696.888... isn't a valid fraction. Hmm, maybe I need to reconsider.Wait, maybe I made a mistake earlier in calculating 384/25 * 49/9. Let me recalculate that. 384 * 49 is 18,816, and 25 * 9 is 225. So, that part is correct. Then, 27 divided by (18,816/225) is 27 * (225/18,816) = 6,075 / 18,816. Simplifying that, I get 2,025 / 6,272, which doesn't reduce further.But the answer choices don't have 2,025/6,272. They have options like 225/696, which is exactly 2,025/6,272 divided by 9. Wait, but 6,272 divided by 9 is not a whole number, so maybe I did something wrong earlier.Let me go back to the equation: 27 = (384/25) * (Q / (9/49)). Maybe I should handle the fractions more carefully. So, Q divided by (9/49) is Q * (49/9). So, the equation is 27 = (384/25) * (49/9) * Q.Let me compute (384/25) * (49/9). 384 divided by 9 is 42.666..., and 49 divided by 25 is 1.96. So, 42.666... * 1.96 is approximately 83.6. But I need to keep it as a fraction.Alternatively, maybe I can simplify before multiplying. 384 and 9 have a common factor of 3. 384 divided by 3 is 128, and 9 divided by 3 is 3. So, now it's (128/25) * (49/3). 128 and 3 have no common factors, and 49 and 25 have no common factors. So, multiplying numerators: 128 * 49 = 6,272, and denominators: 25 * 3 = 75. So, it's 6,272/75.Wait, that's different from what I had before. Earlier, I had 18,816/225, but now I have 6,272/75. Wait, 6,272/75 is the same as 18,816/225 because 6,272 * 3 = 18,816 and 75 * 3 = 225. So, both are equivalent.So, going back, 27 = (6,272/75) * Q. To solve for Q, divide both sides by (6,272/75), which is the same as multiplying by 75/6,272. So, Q = 27 * (75/6,272).Let me compute that. 27 * 75 is 2,025, and the denominator is 6,272. So, Q = 2,025 / 6,272. Hmm, same as before.Wait, but looking at the answer choices, option B is 225/696. Let me see if 2,025/6,272 can be simplified to 225/696. Let's divide numerator and denominator by 9: 2,025 ÷ 9 = 225, and 6,272 ÷ 9 = 696.888..., which isn't a whole number. So, that doesn't work.Wait, maybe I made a mistake in my initial setup. Let me double-check. The relationship is P = k * (Q/U). When P = 12, Q = 1/2, U = 16/25. So, 12 = k * (1/2) / (16/25). That simplifies to 12 = k * (25/32). So, k = 12 * (32/25) = 384/25. That seems correct.Then, when P = 27 and U = 9/49, we have 27 = (384/25) * (Q / (9/49)). So, Q / (9/49) is Q * (49/9). So, 27 = (384/25) * (49/9) * Q. So, 384/25 * 49/9 = (384*49)/(25*9) = 18,816/225. So, 27 = (18,816/225) * Q. Therefore, Q = 27 * (225/18,816) = 6,075/18,816 = 2,025/6,272.Hmm, maybe I need to check if 2,025 and 6,272 have any common factors. Let's see, 2,025 is 25 * 81, which is 5^2 * 3^4. 6,272 is 64 * 98, which is 2^6 * 7^2. So, no common factors. Therefore, 2,025/6,272 is the simplest form.But none of the answer choices match that. Wait, let me look at the options again: (A) 200/699, (B) 225/696, (C) 250/693, (D) 300/691, (E) 325/688.Hmm, none of these are 2,025/6,272. Wait, maybe I made a mistake in simplifying. Let me try another approach. Maybe I can reduce the fraction 6,075/18,816.Divide numerator and denominator by 3: 6,075 ÷ 3 = 2,025; 18,816 ÷ 3 = 6,272. So, same as before. Divide by 3 again: 2,025 ÷ 3 = 675; 6,272 ÷ 3 = 2,090.666..., which isn't a whole number. So, can't divide by 3 again.Divide by 5: 675 ÷ 5 = 135; 2,090.666... ÷ 5 isn't a whole number. So, no. Maybe 25? 675 ÷ 25 = 27; 2,090.666... ÷ 25 isn't a whole number. Hmm.Wait, maybe I need to see if 2,025 and 6,272 have a common factor of 3 again. 2,025 ÷ 3 = 675; 6,272 ÷ 3 ≈ 2,090.666..., which isn't a whole number. So, no.Wait, maybe I made a mistake in the initial calculation of 384/25 * 49/9. Let me compute that again. 384 * 49: 384 * 50 = 19,200; subtract 384 to get 19,200 - 384 = 18,816. So, numerator is 18,816. Denominator is 25 * 9 = 225. So, 18,816/225. Then, 27 divided by (18,816/225) is 27 * (225/18,816) = 6,075/18,816.Wait, maybe I can divide numerator and denominator by 27. 6,075 ÷ 27 = 225; 18,816 ÷ 27 = 696.888..., which isn't a whole number. So, that doesn't work.Wait, but 18,816 ÷ 27: 27 * 696 = 18,816 - let me check. 27 * 700 = 18,900, which is 84 more than 18,816. So, 700 - (84/27) = 700 - 3.111... = 696.888... So, yes, 27 * 696.888... = 18,816. So, 6,075/18,816 = 225/696.888..., which isn't a whole number.Wait, but the answer choices have 225/696 as option B. Maybe the question expects us to approximate or consider that 696.888... is approximately 696. So, maybe they rounded it. But that seems unlikely.Alternatively, maybe I made a mistake in the initial setup. Let me try another approach. Maybe I can set up the proportion directly.Since P varies directly as Q and inversely as U, the relationship can be written as P1 * U1 / Q1 = P2 * U2 / Q2. So, (P1 * U1)/Q1 = (P2 * U2)/Q2.Plugging in the values: (12 * (16/25)) / (1/2) = (27 * (9/49)) / Q2.Let me compute the left side first: 12 * (16/25) = 192/25. Then, divide by (1/2) is the same as multiplying by 2, so 192/25 * 2 = 384/25.So, 384/25 = (27 * (9/49)) / Q2.Compute the right side numerator: 27 * (9/49) = 243/49.So, 384/25 = (243/49) / Q2.To solve for Q2, we can write Q2 = (243/49) / (384/25) = (243/49) * (25/384).Multiply numerators: 243 * 25 = 6,075.Multiply denominators: 49 * 384 = 18,816.So, Q2 = 6,075 / 18,816, which simplifies to 2,025 / 6,272, same as before.Hmm, so it seems like I keep getting 2,025/6,272, which isn't one of the answer choices. Wait, maybe I need to simplify it differently. Let me see, 2,025 divided by 9 is 225, and 6,272 divided by 9 is approximately 696.888..., which isn't a whole number. So, maybe the answer is supposed to be 225/696, which is option B, even though it's not exact.Alternatively, maybe I made a mistake in the initial calculation of k. Let me check that again. P = k * Q/U. So, 12 = k * (1/2) / (16/25). So, (1/2) / (16/25) = (1/2) * (25/16) = 25/32. So, 12 = k * (25/32). Therefore, k = 12 * (32/25) = 384/25. That seems correct.Wait, maybe I can express 2,025/6,272 in terms of the answer choices. Let me see, 2,025 divided by 9 is 225, and 6,272 divided by 9 is approximately 696.888..., which is close to 696. So, maybe the answer is 225/696, which is option B.Alternatively, maybe I need to check my arithmetic again. Let me compute 27 * (225/18,816). 27 * 225 = 6,075, and 6,075 / 18,816 = 0.322... Let me see what 225/696 is approximately. 225 ÷ 696 ≈ 0.323. So, they are very close. So, maybe due to rounding, the answer is 225/696.Alternatively, maybe I made a mistake in simplifying the fraction. Let me see, 6,075/18,816. Let's divide numerator and denominator by 27: 6,075 ÷ 27 = 225; 18,816 ÷ 27 = 696.888..., which isn't a whole number. So, that doesn't work.Wait, maybe I can divide numerator and denominator by 3: 6,075 ÷ 3 = 2,025; 18,816 ÷ 3 = 6,272. Then, divide by 3 again: 2,025 ÷ 3 = 675; 6,272 ÷ 3 ≈ 2,090.666..., which isn't a whole number. So, no.Wait, maybe I can divide numerator and denominator by 5: 6,075 ÷ 5 = 1,215; 18,816 ÷ 5 = 3,763.2, which isn't a whole number. So, no.Wait, maybe I can divide numerator and denominator by 15: 6,075 ÷ 15 = 405; 18,816 ÷ 15 = 1,254.4, which isn't a whole number. So, no.Hmm, I'm stuck. I keep getting 2,025/6,272, which isn't one of the answer choices. But the closest is 225/696, which is option B. Maybe I need to accept that and choose B.Alternatively, maybe I made a mistake in the initial setup. Let me try another approach. Maybe I can set up the proportion as P1/Q1 = P2/Q2 * (U2/U1). Wait, no, that doesn't seem right.Wait, since P varies directly as Q and inversely as U, the relationship is P = kQ/U. So, P1U1/Q1 = P2U2/Q2. So, (12 * (16/25)) / (1/2) = (27 * (9/49)) / Q2.Compute left side: 12 * (16/25) = 192/25. Divided by (1/2) is 192/25 * 2 = 384/25.Right side: 27 * (9/49) = 243/49. So, 384/25 = (243/49)/Q2. So, Q2 = (243/49) / (384/25) = (243/49) * (25/384) = (243*25)/(49*384) = 6,075/18,816.Same result as before. So, I think I have to go with 2,025/6,272, but since that's not an option, maybe I made a mistake in the problem setup.Wait, maybe I misread the problem. Let me check again. P varies directly as Q and inversely as U. When P = 12 and U = 16/25, Q = 1/2. Find Q when P = 27 and U = 9/49.Yes, that's what I did. So, maybe the answer is supposed to be 225/696, which is option B, even though my calculation gives a slightly different value. Maybe I need to accept that and choose B.

answer：I need to simplify the expression (7y - 3x + 8 + 2y^2 - x + 12).First, I'll identify the like terms:- The terms with (x) are (-3x) and (-x).- The terms with (y) are (7y).- The quadratic term is (2y^2).- The constant terms are (8) and (12).Next, I'll combine the like terms:- Combining (-3x) and (-x) gives (-4x).- Combining (8) and (12) gives (20).Now, I'll rewrite the expression with the combined terms:(2y^2 + 7y - 4x + 20).This is the simplified form of the original expression.

answer：Okay, so I have this problem where I'm given two expressions: x is equal to the square root of 3 plus the square root of 2, and y is equal to the square root of 3 minus the square root of 2. I need to find the value of the algebraic expression xy squared minus x squared y. Hmm, that sounds a bit complicated, but let me try to break it down step by step.First, let me write down what I know:x = √3 + √2y = √3 - √2And the expression I need to evaluate is:xy² - x²yAlright, so maybe I can factor this expression somehow to make it simpler. Let me see... If I look at xy² - x²y, I notice that both terms have an x and a y in them. Maybe I can factor out an xy?Let me try that:xy² - x²y = xy(y) - xy(x) = xy(y - x)Oh, that's nice! So now the expression simplifies to xy times (y minus x). That seems easier to handle. So now I have:xy(y - x)Now, I need to compute this product. Let's compute each part step by step.First, let's compute xy. That's x multiplied by y. Given that x is √3 + √2 and y is √3 - √2, I can use the difference of squares formula here. Remember, (a + b)(a - b) = a² - b². So applying that here:xy = (√3 + √2)(√3 - √2) = (√3)² - (√2)² = 3 - 2 = 1Oh, that's neat! So xy is just 1. That simplifies things a lot.Now, the expression becomes:xy(y - x) = 1 * (y - x) = y - xSo now I just need to compute y minus x. Let's do that.y - x = (√3 - √2) - (√3 + √2) = √3 - √2 - √3 - √2Let me simplify that:√3 - √3 cancels out, right? So that leaves me with -√2 - √2, which is -2√2.So putting it all together:xy² - x²y = xy(y - x) = 1 * (y - x) = -2√2Wait, that seems straightforward. But let me double-check my steps to make sure I didn't make any mistakes.First, I factored the expression correctly: xy² - x²y = xy(y - x). That seems right.Then, I calculated xy using the difference of squares: (√3 + √2)(√3 - √2) = 3 - 2 = 1. That checks out.Next, I computed y - x: (√3 - √2) - (√3 + √2) = -2√2. Yep, that's correct.So, multiplying xy (which is 1) by (y - x) gives me -2√2. So the final answer should be -2√2.But just to be thorough, maybe I can compute the original expression without factoring to see if I get the same result.Let's compute xy² and x²y separately and then subtract them.First, compute y²:y = √3 - √2y² = (√3 - √2)² = (√3)² - 2*√3*√2 + (√2)² = 3 - 2√6 + 2 = 5 - 2√6Now, compute xy²:x = √3 + √2xy² = (√3 + √2)(5 - 2√6)Let's multiply this out:= √3*5 + √3*(-2√6) + √2*5 + √2*(-2√6)= 5√3 - 2√18 + 5√2 - 2√12Simplify the radicals:√18 = √(9*2) = 3√2√12 = √(4*3) = 2√3So substituting back:= 5√3 - 2*(3√2) + 5√2 - 2*(2√3)= 5√3 - 6√2 + 5√2 - 4√3Combine like terms:5√3 - 4√3 = √3-6√2 + 5√2 = -√2So, xy² = √3 - √2Now, let's compute x²y.First, compute x²:x = √3 + √2x² = (√3 + √2)² = (√3)² + 2*√3*√2 + (√2)² = 3 + 2√6 + 2 = 5 + 2√6Now, compute x²y:x²y = (5 + 2√6)(√3 - √2)Multiply this out:= 5*√3 + 5*(-√2) + 2√6*√3 + 2√6*(-√2)= 5√3 - 5√2 + 2√18 - 2√12Simplify the radicals:√18 = 3√2√12 = 2√3Substituting back:= 5√3 - 5√2 + 2*(3√2) - 2*(2√3)= 5√3 - 5√2 + 6√2 - 4√3Combine like terms:5√3 - 4√3 = √3-5√2 + 6√2 = √2So, x²y = √3 + √2Now, subtract x²y from xy²:xy² - x²y = (√3 - √2) - (√3 + √2) = √3 - √2 - √3 - √2 = -2√2Okay, so that confirms my earlier result. Whether I factor the expression first or compute each term separately, I end up with the same answer: -2√2. That makes me more confident that this is the correct solution.I think I understand the problem now. By recognizing that the expression can be factored, I could simplify the computation significantly. It's also good to verify the result by expanding everything out, just to make sure there were no mistakes in the factoring step. Both methods led me to the same conclusion, so I feel pretty certain about the answer.Another thing I noticed is that x and y are conjugates of each other. That is, x = √3 + √2 and y = √3 - √2. This is useful because their product is a rational number, as we saw earlier: xy = 1. This kind of relationship often helps in simplifying expressions, especially when dealing with radicals.Also, when I computed y - x, I saw that the √3 terms canceled out, leaving me with a simple expression involving only √2. That was a nice simplification and shows how the structure of x and y can lead to cancellation of certain terms.In summary, the key steps were:1. Factor the original expression to simplify it.2. Recognize that x and y are conjugates, so their product is a rational number.3. Compute the necessary products and differences step by step.4. Verify the result by expanding the original expression without factoring.This problem was a good exercise in algebraic manipulation, factoring, and working with radicals. It also reinforced the importance of checking one's work through alternative methods to ensure accuracy.