💡Okay, so I have this problem: "For how many four-digit positive integers is the sum of the digits equal to 9?" Hmm, let me try to figure this out step by step. First, I know that a four-digit number ranges from 1000 to 9999. So, the first digit can't be zero; it has to be at least 1. The other three digits can be anything from 0 to 9. The problem is asking for how many such numbers have digits that add up to 9. Let me denote the four-digit number as ( abcd ), where ( a ), ( b ), ( c ), and ( d ) are its digits. So, ( a ) is the thousands place, ( b ) is the hundreds, ( c ) is the tens, and ( d ) is the ones place. The condition given is that ( a + b + c + d = 9 ). But wait, since ( a ) is the first digit, it can't be zero. So, ( a ) must be at least 1. That means ( a ) can be 1 through 9, but since the total sum is 9, ( a ) can't be more than 9. So, ( a ) is between 1 and 9, and ( b ), ( c ), ( d ) are between 0 and 9. I remember that problems like this, where we need to find the number of non-negative integer solutions to an equation, can be solved using the stars and bars method. But in this case, since ( a ) has to be at least 1, I might need to adjust for that. Let me think. If ( a ) has to be at least 1, maybe I can subtract 1 from ( a ) to make it a non-negative variable. Let me define ( a' = a - 1 ). Then, ( a' ) will be at least 0, and the equation becomes ( a' + 1 + b + c + d = 9 ), which simplifies to ( a' + b + c + d = 8 ). Now, all variables ( a' ), ( b ), ( c ), and ( d ) are non-negative integers, and I need to find the number of solutions to this equation. This is a classic stars and bars problem where I have to distribute 8 identical "stars" (the total sum) into 4 "bins" (the variables). The formula for the number of non-negative integer solutions is given by the combination formula ( binom{n + k - 1}{k - 1} ), where ( n ) is the total and ( k ) is the number of variables. In this case, ( n = 8 ) and ( k = 4 ). So, the number of solutions should be ( binom{8 + 4 - 1}{4 - 1} = binom{11}{3} ). Let me calculate ( binom{11}{3} ). The combination formula is ( binom{n}{r} = frac{n!}{r!(n - r)!} ). So, ( binom{11}{3} = frac{11!}{3! cdot 8!} ). Calculating that, ( 11! = 39916800 ), ( 3! = 6 ), and ( 8! = 40320 ). So, ( frac{39916800}{6 cdot 40320} = frac{39916800}{241920} ). Let me do the division: 39916800 divided by 241920. Hmm, 241920 times 165 is 39916800 because 241920 * 100 = 24,192,000; 241920 * 60 = 14,515,200; 241920 * 5 = 1,209,600. Adding those together: 24,192,000 + 14,515,200 = 38,707,200; plus 1,209,600 is 39,916,800. So, yes, 241920 * 165 = 39,916,800. Therefore, ( binom{11}{3} = 165 ). Wait, but I should make sure that none of the digits exceed 9, right? Because in the stars and bars method, we assume that each variable can take any non-negative integer value, but in reality, each digit can't be more than 9. Let me check if any of the digits could exceed 9 in this scenario. The total sum is 8 after adjusting for ( a' ). Since each digit is a non-negative integer, the maximum any single digit could be is 8, which is less than 9. So, none of the digits ( a' ), ( b ), ( c ), or ( d ) can exceed 8, which is within the 0-9 range. Therefore, there's no need to subtract any cases where a digit exceeds 9 because it's impossible in this situation. So, the number of four-digit positive integers where the sum of the digits is 9 is 165.**Final Answer**The number of four-digit positive integers with a digit sum of 9 is boxed{165}.

💡First, I need to determine the number of days the contractor was absent. I'll start by defining two variables: let ( x ) represent the number of days the contractor worked, and ( y ) represent the number of days he was absent.Since the total duration of the engagement is 30 days, I can write the first equation as:[ x + y = 30 ]Next, considering the payment structure, the contractor earns Rs. 25 for each day worked and is fined Rs. 7.50 for each day absent. The total amount he received is Rs. 425. This gives me the second equation:[ 25x - 7.5y = 425 ]Now, I have a system of two equations:1. ( x + y = 30 )2. ( 25x - 7.5y = 425 )To solve this system, I'll use the substitution method. From the first equation, I can express ( y ) in terms of ( x ):[ y = 30 - x ]Substituting this expression for ( y ) into the second equation:[ 25x - 7.5(30 - x) = 425 ]Expanding and simplifying:[ 25x - 225 + 7.5x = 425 ][ 32.5x - 225 = 425 ][ 32.5x = 650 ][ x = frac{650}{32.5} ][ x = 20 ]So, the contractor worked for 20 days. To find the number of days he was absent:[ y = 30 - x ][ y = 30 - 20 ][ y = 10 ]Therefore, the contractor was absent for 10 days.

💡Okay, so I have these three functions to find the derivatives of. Let me take them one by one and think through each step carefully. I remember from my calculus class that derivatives involve finding the rate at which a function changes, and there are specific rules like the product rule, quotient rule, and chain rule that I might need to use here.Starting with the first function: ( y = (x + 1)^2(x - 1) ). Hmm, this looks like a product of two functions: one is ( (x + 1)^2 ) and the other is ( (x - 1) ). So, I think I should use the product rule here. The product rule says that if you have two functions multiplied together, say ( u(x) ) and ( v(x) ), then the derivative ( y' ) is ( u'(x)v(x) + u(x)v'(x) ).Let me assign ( u(x) = (x + 1)^2 ) and ( v(x) = (x - 1) ). Now, I need to find ( u'(x) ) and ( v'(x) ). For ( u(x) = (x + 1)^2 ), I can use the chain rule. The chain rule is for composite functions, so here, the outer function is something squared, and the inner function is ( x + 1 ). The derivative of the outer function is ( 2(x + 1) ), and the derivative of the inner function is 1. So, multiplying them together, ( u'(x) = 2(x + 1) times 1 = 2(x + 1) ).For ( v(x) = (x - 1) ), the derivative is straightforward. The derivative of ( x ) is 1, and the derivative of -1 is 0, so ( v'(x) = 1 ).Now, applying the product rule: ( y' = u'(x)v(x) + u(x)v'(x) ). Plugging in the values, that's ( y' = 2(x + 1)(x - 1) + (x + 1)^2 times 1 ).Let me simplify this expression. First, expand ( 2(x + 1)(x - 1) ). That's ( 2[(x)(x) + x(-1) + 1(x) + 1(-1)] ) which simplifies to ( 2[x^2 - x + x - 1] ). The middle terms cancel out, so it becomes ( 2(x^2 - 1) ) which is ( 2x^2 - 2 ).Next, expand ( (x + 1)^2 ). That's ( x^2 + 2x + 1 ).Now, add the two results together: ( 2x^2 - 2 + x^2 + 2x + 1 ). Combine like terms: ( 2x^2 + x^2 = 3x^2 ), ( 2x ) stays as is, and ( -2 + 1 = -1 ). So, the derivative simplifies to ( 3x^2 + 2x - 1 ).Wait, that seems straightforward. Let me double-check by expanding the original function before differentiating. If I expand ( (x + 1)^2(x - 1) ), it should give me a polynomial which I can then differentiate term by term.Expanding ( (x + 1)^2 ) gives ( x^2 + 2x + 1 ). Then multiplying by ( (x - 1) ):( (x^2 + 2x + 1)(x - 1) = x^3 - x^2 + 2x^2 - 2x + x - 1 ). Combining like terms: ( x^3 + ( -x^2 + 2x^2 ) + ( -2x + x ) - 1 ), which simplifies to ( x^3 + x^2 - x - 1 ).Now, taking the derivative of this polynomial: ( d/dx (x^3 + x^2 - x - 1) = 3x^2 + 2x - 1 - 0 ), which is the same as before. So, that confirms my earlier result. Good, so the derivative is ( 3x^2 + 2x - 1 ).Moving on to the second function: ( y = x^2 sin x ). Again, this is a product of two functions: ( x^2 ) and ( sin x ). So, I'll use the product rule here as well.Let me denote ( u(x) = x^2 ) and ( v(x) = sin x ). Then, ( u'(x) = 2x ) and ( v'(x) = cos x ).Applying the product rule: ( y' = u'(x)v(x) + u(x)v'(x) = 2x sin x + x^2 cos x ).That seems straightforward. Let me think if there's another way to approach this. Maybe using the definition of the derivative? But that would involve limits and might be more complicated. The product rule seems the most efficient here.Alternatively, I could also consider if there's a trigonometric identity that could simplify ( x^2 sin x ), but I don't think that's necessary here. The product rule gives a direct answer. So, I think ( y' = 2x sin x + x^2 cos x ) is correct.For the third function: ( y = frac{e^{x} + 1}{e^{x} - 1} ). This is a quotient of two functions, so I should use the quotient rule. The quotient rule states that if ( y = frac{u(x)}{v(x)} ), then ( y' = frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ).Let me assign ( u(x) = e^{x} + 1 ) and ( v(x) = e^{x} - 1 ). Then, I need to find ( u'(x) ) and ( v'(x) ).The derivative of ( u(x) = e^{x} + 1 ) is ( u'(x) = e^{x} + 0 = e^{x} ).Similarly, the derivative of ( v(x) = e^{x} - 1 ) is ( v'(x) = e^{x} - 0 = e^{x} ).Now, applying the quotient rule: ( y' = frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ).Plugging in the values:( y' = frac{e^{x}(e^{x} - 1) - (e^{x} + 1)e^{x}}{(e^{x} - 1)^2} ).Let me simplify the numerator step by step.First, expand ( e^{x}(e^{x} - 1) ):( e^{x} times e^{x} = e^{2x} ) and ( e^{x} times (-1) = -e^{x} ). So, that term becomes ( e^{2x} - e^{x} ).Next, expand ( (e^{x} + 1)e^{x} ):( e^{x} times e^{x} = e^{2x} ) and ( 1 times e^{x} = e^{x} ). So, that term becomes ( e^{2x} + e^{x} ).Now, subtract the second expansion from the first:( (e^{2x} - e^{x}) - (e^{2x} + e^{x}) = e^{2x} - e^{x} - e^{2x} - e^{x} ).Simplify term by term:( e^{2x} - e^{2x} = 0 ),( -e^{x} - e^{x} = -2e^{x} ).So, the numerator simplifies to ( -2e^{x} ).Therefore, the derivative ( y' ) is ( frac{-2e^{x}}{(e^{x} - 1)^2} ).Let me double-check my steps to make sure I didn't make a mistake. Starting from the quotient rule, I correctly identified ( u(x) ) and ( v(x) ), found their derivatives, and substituted into the formula. The expansion of the numerator seems correct: subtracting the two products gives ( -2e^{x} ). So, the final derivative should be ( frac{-2e^{x}}{(e^{x} - 1)^2} ).Alternatively, I could also consider simplifying the original function before differentiating, but I don't see an obvious simplification. So, the quotient rule was the right approach here.In summary, for each function, I identified the appropriate rule (product or quotient), applied it step by step, and then simplified the result. I also double-checked my work by expanding the original function for the first problem, which confirmed my answer. For the other two, the product and quotient rules gave direct results that seem correct upon review.**Final Answer**(1) The derivative is boxed{3x^2 + 2x - 1}.(2) The derivative is boxed{2x sin x + x^2 cos x}.(3) The derivative is boxed{dfrac{-2e^{x}}{(e^{x} - 1)^2}}.

💡Okay, so I have this chemistry problem here about finding out how many moles of hydrochloric acid are used when combining 2 moles of amyl alcohol to produce 2 moles of 1-chloro-2,2-dimethylpropane. Hmm, let me try to break this down step by step.First, I need to understand what the reactants and products are. Amyl alcohol, also known as pentanol or 2-methylbutan-2-ol, is the alcohol in question. Hydrochloric acid (HCl) is the acid reacting with it. The product is 1-chloro-2,2-dimethylpropane, which I think is also called isopropyl chloride or something similar.So, the reaction is between an alcohol and an acid, which typically results in an esterification or substitution reaction. In this case, since we're replacing the hydroxyl group (-OH) of the alcohol with a chlorine atom from HCl, it's a substitution reaction.Let me write down the general chemical equation for this reaction:C5H11OH + HCl → C5H11Cl + H2OThis looks like a typical acid-catalyzed substitution where the -OH group is replaced by Cl. The stoichiometry here seems to be 1:1:1:1, meaning one mole of amyl alcohol reacts with one mole of HCl to produce one mole of the chlorinated product and one mole of water.Given that, if I start with 2 moles of amyl alcohol, and the reaction produces 2 moles of 1-chloro-2,2-dimethylpropane, then according to the stoichiometry, I would need 2 moles of HCl to react completely with the 2 moles of amyl alcohol.But wait, I should double-check if there are any side reactions or if the reaction goes to completion. In real scenarios, reactions might not go to 100% completion, but since the problem states that 2 moles of the product are formed, I can assume that the reaction goes to completion, and all the amyl alcohol is converted into the chlorinated product.Also, I need to make sure that the molar ratios are correct. For every mole of amyl alcohol, one mole of HCl is required. So, 2 moles of amyl alcohol would require 2 moles of HCl.Is there anything else I need to consider? Maybe the state of the reactants or the conditions under which the reaction occurs? The problem doesn't specify any additional details, so I think it's safe to proceed with the stoichiometric calculation.In summary, based on the balanced chemical equation and the given amounts, 2 moles of amyl alcohol would react with 2 moles of hydrochloric acid to produce 2 moles of 1-chloro-2,2-dimethylpropane.

💡Okay, so I have this infinite series to evaluate: 1/3 + 2/3² + 3/3³ + ... + k/3^k + ... and so on. Hmm, infinite series can be tricky, but I remember that some of them can be summed up using specific formulas. Let me think about how to approach this.First, I notice that each term in the series has the form k divided by 3 to the power of k. So, it's like k/(3^k). That makes me think of arithmetic-geometric series because each term is a product of an arithmetic sequence (k) and a geometric sequence (1/3^k). I remember that there's a formula for the sum of such series.Let me denote the sum as S. So,S = 1/3 + 2/3² + 3/3³ + 4/3⁴ + ... I think a common technique to solve this is to multiply both sides by the common ratio of the geometric part, which in this case is 1/3. Let me try that.Multiplying S by 1/3:(1/3)S = 1/3² + 2/3³ + 3/3⁴ + 4/3⁵ + ... Now, if I subtract this new series from the original series S, maybe some terms will cancel out. Let's write that out:S - (1/3)S = (1/3 + 2/3² + 3/3³ + 4/3⁴ + ...) - (1/3² + 2/3³ + 3/3⁴ + 4/3⁵ + ...)Let me compute the left side first:S - (1/3)S = (2/3)SNow, let's look at the right side. When I subtract the two series term by term, most of the terms should cancel out except for the first few. Let's see:First term: 1/3 remains because there's nothing to subtract from it in the second series.Second term: 2/3² - 1/3² = (2 - 1)/3² = 1/3²Third term: 3/3³ - 2/3³ = (3 - 2)/3³ = 1/3³Fourth term: 4/3⁴ - 3/3⁴ = (4 - 3)/3⁴ = 1/3⁴And so on...So, the right side becomes:1/3 + 1/3² + 1/3³ + 1/3⁴ + ... This is a geometric series where each term is (1/3)^n starting from n=1 to infinity. I remember that the sum of an infinite geometric series is a/(1 - r), where a is the first term and r is the common ratio, provided that |r| < 1.In this case, a = 1/3 and r = 1/3. So, plugging into the formula:Sum = (1/3) / (1 - 1/3) = (1/3) / (2/3) = (1/3) * (3/2) = 1/2So, the right side of the equation is 1/2. Therefore, we have:(2/3)S = 1/2To solve for S, multiply both sides by 3/2:S = (1/2) * (3/2) = 3/4Wait, that doesn't seem right. Earlier, I thought the sum was 3/2. Did I make a mistake somewhere?Let me double-check my steps. I had:S = 1/3 + 2/3² + 3/3³ + ...(1/3)S = 1/3² + 2/3³ + 3/3⁴ + ...Subtracting: S - (1/3)S = (2/3)S = 1/3 + 1/3² + 1/3³ + ... Yes, that's correct. The subtraction gives a geometric series starting at 1/3 with ratio 1/3.Calculating the sum: (1/3) / (1 - 1/3) = (1/3)/(2/3) = 1/2. So, (2/3)S = 1/2, which leads to S = (1/2)*(3/2) = 3/4.But wait, I thought the sum was 3/2. Maybe I confused it with another series. Let me verify with a different approach.Another way to find the sum of k/(3^k) is to use the formula for the sum of an infinite arithmetic-geometric series. The general formula is:Sum = a / (1 - r)^2where a is the first term of the arithmetic sequence and r is the common ratio of the geometric sequence.In this case, a = 1 and r = 1/3. Plugging into the formula:Sum = 1 / (1 - 1/3)^2 = 1 / (2/3)^2 = 1 / (4/9) = 9/4 = 2.25Wait, that's 9/4, which is 2.25, not 3/4 or 3/2. Now I'm confused. Which one is correct?Let me check both methods again.First method:S = 1/3 + 2/3² + 3/3³ + ...(1/3)S = 1/3² + 2/3³ + 3/3⁴ + ...Subtracting: (2/3)S = 1/3 + 1/3² + 1/3³ + ... = (1/3)/(1 - 1/3) = 1/2Thus, S = (1/2)*(3/2) = 3/4Second method:Using the formula for arithmetic-geometric series: Sum = a / (1 - r)^2 = 1 / (1 - 1/3)^2 = 1 / (4/9) = 9/4These two results are conflicting. There must be a mistake in one of the methods.Wait, maybe the formula I used in the second method is incorrect. Let me recall the correct formula for the sum of k*r^k from k=1 to infinity.I think the correct formula is r / (1 - r)^2. Let me verify that.Yes, the sum from k=1 to infinity of k*r^k is r / (1 - r)^2.So, in this case, r = 1/3. Plugging in:Sum = (1/3) / (1 - 1/3)^2 = (1/3) / (2/3)^2 = (1/3) / (4/9) = (1/3)*(9/4) = 3/4Ah, so the correct formula is r / (1 - r)^2, not a / (1 - r)^2. I must have misremembered the formula earlier.Therefore, the first method was correct, and the second method had an error in the formula. So, the sum S is indeed 3/4.But wait, earlier I thought the answer was 3/2. Maybe I confused it with another problem. Let me check with another approach.Let me compute partial sums numerically to see what value they approach.Compute S_n = sum from k=1 to n of k/(3^k)Let's compute S_1 = 1/3 ≈ 0.3333S_2 = 1/3 + 2/9 ≈ 0.3333 + 0.2222 ≈ 0.5555S_3 = 0.5555 + 3/27 ≈ 0.5555 + 0.1111 ≈ 0.6666S_4 = 0.6666 + 4/81 ≈ 0.6666 + 0.0494 ≈ 0.7160S_5 = 0.7160 + 5/243 ≈ 0.7160 + 0.0205 ≈ 0.7365S_6 = 0.7365 + 6/729 ≈ 0.7365 + 0.0082 ≈ 0.7447S_7 = 0.7447 + 7/2187 ≈ 0.7447 + 0.0032 ≈ 0.7479S_8 = 0.7479 + 8/6561 ≈ 0.7479 + 0.0012 ≈ 0.7491S_9 = 0.7491 + 9/19683 ≈ 0.7491 + 0.000457 ≈ 0.749557S_10 ≈ 0.749557 + 10/59049 ≈ 0.749557 + 0.000169 ≈ 0.749726So, as n increases, S_n approaches approximately 0.75, which is 3/4. Therefore, the sum S is indeed 3/4.Wait, but earlier I thought the answer was 3/2. Maybe I confused it with another series where the common ratio was different. Let me confirm.Yes, for the series sum from k=1 to infinity of k/(2^k), the sum is 2. That's where the 3/2 might have come from if I confused the ratio. But in this case, with ratio 1/3, the sum is 3/4.So, to summarize, using the method of subtracting the series multiplied by the common ratio, I derived that S = 3/4. Additionally, computing partial sums numerically supports this result. Therefore, the sum of the series is 3/4.But wait, in the initial problem statement, the user wrote:"Evaluate the sum [1/3 + 2/3² + 3/3³ + ... + k/3^k + ...]"And in the thought process, I arrived at 3/4. However, in the initial assistant's response, it concluded S = 3/2. That seems contradictory.Let me double-check the assistant's solution.The assistant wrote:Let the sum of the series be S.S = 1/3 + 2/3² + 3/3³ + 4/3⁴ + ...(1/3)S = 1/3² + 2/3³ + 3/3⁴ + ...Subtracting:(2/3)S = 1/3 + 1/9 + 1/27 + ... This is a geometric series with a = 1/3 and r = 1/3.Sum = (1/3)/(1 - 1/3) = (1/3)/(2/3) = 1/2Thus, (2/3)S = 1/2 => S = (1/2)*(3/2) = 3/4Wait, that's exactly what I did and arrived at 3/4. But in the initial problem, the user wrote:"Evaluate the sum [1/3 + 2/3² + 3/3³ + ... + k/3^k + ...]"And the assistant's initial response concluded S = 3/2. That seems incorrect because both methods I used and the numerical approximation point to 3/4.Perhaps there was a mistake in the assistant's initial response. Let me check the assistant's steps again.The assistant wrote:S = 1/3 + 2/3² + 3/3³ + 4/3⁴ + ...(1/3)S = 1/3² + 2/3³ + 3/3⁴ + ...Subtracting:(2/3)S = 1/3 + 1/9 + 1/27 + ... Sum of geometric series: (1/3)/(1 - 1/3) = 1/2Thus, S = (1/2)*(3/2) = 3/4Wait, that's correct. So, the assistant's initial response was correct, arriving at 3/4. But in the problem statement, the user wrote:"Evaluate the sum [1/3 + 2/3² + 3/3³ + ... + k/3^k + ...]"And the assistant's initial response concluded S = 3/2, which is incorrect. There must have been a mistake in the initial response. Let me correct that.Therefore, the correct sum is 3/4, not 3/2. The initial assistant's response had an error in the calculation, arriving at 3/2 instead of 3/4. The correct approach, as shown in the thought process, leads to S = 3/4.So, to conclude, the sum of the series is 3/4.

💡First, I need to determine Kamal's current age based on the given information.I'll start by defining variables for Kamal's and his son's current ages. Let ( K ) represent Kamal's current age and ( S ) represent his son's current age.From the first statement, 8 years ago, Kamal was 4 times as old as his son. This gives me the equation:[K - 8 = 4(S - 8)]From the second statement, after 8 years, Kamal will be twice as old as his son. This provides another equation:[K + 8 = 2(S + 8)]Now, I have a system of two equations with two variables. I'll solve this system to find the values of ( K ) and ( S ).First, I'll express ( S ) in terms of ( K ) using the first equation:[K - 8 = 4(S - 8)][K - 8 = 4S - 32][K = 4S - 24]Next, I'll substitute this expression for ( K ) into the second equation:[4S - 24 + 8 = 2(S + 8)][4S - 16 = 2S + 16][2S = 32][S = 16]Now that I have the son's current age, I'll find Kamal's current age:[K = 4S - 24][K = 4(16) - 24][K = 64 - 24][K = 40]Finally, I'll verify the solution to ensure it satisfies both original conditions.8 years ago:[text{Kamal's age} = 40 - 8 = 32][text{Son's age} = 16 - 8 = 8][32 = 4 times 8 quad text{(True)}]After 8 years:[text{Kamal's age} = 40 + 8 = 48][text{Son's age} = 16 + 8 = 24][48 = 2 times 24 quad text{(True)}]Both conditions are satisfied, confirming that Kamal's current age is 40 years.