answer：First, I need to determine the total number of ways James can choose 2 apples out of the 9 he has. This can be calculated using the combination formula ( binom{9}{2} ), which equals 36.Next, I'll calculate the number of ways to choose 2 green apples from the 4 available. Using the combination formula ( binom{4}{2} ), this equals 6.Finally, to find the probability that both chosen apples are green, I'll divide the number of favorable outcomes (choosing 2 green apples) by the total number of possible outcomes (choosing any 2 apples). This gives ( frac{6}{36} ), which simplifies to ( frac{1}{6} ).

answer：Okay, so I have this equation here: (1 + 2i)z = 3 - i. I need to find the complex number z. Hmm, complex numbers can be a bit tricky, but I think I remember some basics from my math class. Let me try to recall.First, I know that a complex number is usually written as a + bi, where a is the real part and b is the imaginary part. So z should be something like that. The equation is (1 + 2i) multiplied by z equals 3 - i. To find z, I probably need to isolate it on one side of the equation.So, if I have (1 + 2i)z = 3 - i, then z should be equal to (3 - i) divided by (1 + 2i). That makes sense because if I multiply both sides by the reciprocal of (1 + 2i), I should get z by itself. So, z = (3 - i)/(1 + 2i).But dividing complex numbers isn't straightforward like dividing real numbers. I think I need to multiply the numerator and the denominator by the conjugate of the denominator to simplify it. The conjugate of a complex number a + bi is a - bi, right? So, the conjugate of (1 + 2i) should be (1 - 2i).Let me write that down. So, z = (3 - i)/(1 + 2i) multiplied by (1 - 2i)/(1 - 2i). That way, I'm essentially multiplying by 1, which doesn't change the value, but it helps to eliminate the imaginary unit from the denominator.Now, I need to carry out this multiplication. Let's start with the numerator: (3 - i)(1 - 2i). I'll use the distributive property, also known as the FOIL method for multiplying two binomials.First, I'll multiply the First terms: 3 * 1 = 3.Next, the Outer terms: 3 * (-2i) = -6i.Then, the Inner terms: (-i) * 1 = -i.Lastly, the Last terms: (-i) * (-2i) = 2i².Now, I'll add all these together: 3 - 6i - i + 2i².Wait, I remember that i² is equal to -1, so 2i² is 2*(-1) = -2. So, substituting that in, the numerator becomes 3 - 6i - i - 2.Combining like terms: 3 - 2 = 1, and -6i - i = -7i. So, the numerator simplifies to 1 - 7i.Now, let's work on the denominator: (1 + 2i)(1 - 2i). Again, I'll use the distributive property.First terms: 1 * 1 = 1.Outer terms: 1 * (-2i) = -2i.Inner terms: 2i * 1 = 2i.Last terms: 2i * (-2i) = -4i².Adding these together: 1 - 2i + 2i - 4i².Simplifying, the -2i and +2i cancel each other out, leaving 1 - 4i².Again, since i² = -1, this becomes 1 - 4*(-1) = 1 + 4 = 5.So, the denominator simplifies to 5.Putting it all together, z = (1 - 7i)/5.I can separate this into real and imaginary parts: 1/5 - (7/5)i.Let me double-check my work to make sure I didn't make any mistakes. Starting from the beginning:(1 + 2i)z = 3 - iz = (3 - i)/(1 + 2i)Multiply numerator and denominator by (1 - 2i):Numerator: (3 - i)(1 - 2i) = 3*1 + 3*(-2i) + (-i)*1 + (-i)*(-2i) = 3 - 6i - i + 2i² = 3 - 7i - 2 = 1 - 7iDenominator: (1 + 2i)(1 - 2i) = 1*1 + 1*(-2i) + 2i*1 + 2i*(-2i) = 1 - 2i + 2i - 4i² = 1 - 4i² = 1 + 4 = 5So, z = (1 - 7i)/5 = 1/5 - (7/5)iYes, that seems correct. I don't think I made any calculation errors. Maybe I should plug it back into the original equation to verify.Let's compute (1 + 2i) * z, where z = 1/5 - (7/5)i.First, multiply 1 by z: 1*(1/5 - 7/5i) = 1/5 - 7/5iThen, multiply 2i by z: 2i*(1/5 - 7/5i) = (2i)/5 - (14i²)/5Simplify: (2i)/5 - (14*(-1))/5 = (2i)/5 + 14/5Now, add the two results together:1/5 - 7/5i + 14/5 + 2i/5Combine like terms:(1/5 + 14/5) + (-7/5i + 2/5i) = 15/5 + (-5/5i) = 3 - iWhich is exactly the right-hand side of the original equation. So, my solution checks out.I think I've got it right. The complex number z is 1/5 minus 7/5i.**Final Answer**The complex number ( z ) is boxed{frac{1}{5} - frac{7}{5}i}.

answer：Okay, so I have these two vector problems to solve. Let me take them one at a time. Starting with the first one: Given that the magnitude of vector a is 4, the magnitude of vector b is 3, and the angle between them is 60 degrees. I need to find the dot product of (a + 2b) and (a - 3b). Hmm, dot product. I remember that the dot product of two vectors is calculated by multiplying their magnitudes and the cosine of the angle between them. But in this case, it's the dot product of two expressions involving vectors a and b. Let me recall the distributive property of the dot product. So, (a + 2b) · (a - 3b) should expand similarly to how algebraic expressions do. That is, I can distribute each term:(a + 2b) · (a - 3b) = a · a + a · (-3b) + 2b · a + 2b · (-3b)Simplifying each term:a · a is just the magnitude of a squared, so that's |a|², which is 4² = 16.Then, a · (-3b) is -3 times (a · b). Similarly, 2b · a is 2 times (a · b). So, combining these two, we have (-3 + 2)(a · b) = -1(a · b).Next, 2b · (-3b) is -6 times (b · b). And b · b is |b|², which is 3² = 9. So, that term becomes -6 * 9 = -54.Putting it all together:16 - (a · b) - 54Now, I need to find a · b. Since the angle between a and b is 60 degrees, a · b = |a||b|cosθ. Plugging in the values:a · b = 4 * 3 * cos(60°). Cos(60°) is 0.5, so that's 12 * 0.5 = 6.So, substituting back into the expression:16 - 6 - 54 = (16 - 6) - 54 = 10 - 54 = -44.Wait, that seems straightforward. Let me double-check my steps. Expanding the dot product correctly, calculating each term, computing a · b correctly. Yeah, I think that's right. So, the answer to the first part is -44.Moving on to the second problem: Given that (2a - 3b) · (2a + b) = 61, find the angle between a and b. Hmm, okay. So, similar to the first problem, I'll need to expand this dot product.Let me write it out:(2a - 3b) · (2a + b) = 2a · 2a + 2a · b - 3b · 2a - 3b · bWait, no, that's not quite right. Let me correct that. The expansion should be:= 2a · 2a + 2a · b - 3b · 2a - 3b · bWait, actually, that's not the standard expansion. Let me think again. The correct expansion is:(2a - 3b) · (2a + b) = 2a · 2a + 2a · b - 3b · 2a - 3b · bWait, no, that seems off. Let me use the distributive property properly:= 2a · 2a + 2a · b - 3b · 2a - 3b · bWait, actually, that's not correct. The correct expansion is:= (2a) · (2a) + (2a) · b + (-3b) · (2a) + (-3b) · bSo, that's:= 4(a · a) + 2(a · b) - 6(b · a) - 3(b · b)Since the dot product is commutative, a · b = b · a, so we can combine those terms:= 4|a|² + (2 - 6)(a · b) - 3|b|²Simplify:= 4(16) + (-4)(a · b) - 3(9)Because |a| is 4, so |a|² is 16, and |b| is 3, so |b|² is 9.Calculating each term:4 * 16 = 64-4(a · b) remains as is.3 * 9 = 27, so -3 * 9 = -27.Putting it all together:64 - 4(a · b) - 27 = (64 - 27) - 4(a · b) = 37 - 4(a · b)We are told that this equals 61:37 - 4(a · b) = 61So, solving for a · b:-4(a · b) = 61 - 37 = 24Divide both sides by -4:a · b = 24 / (-4) = -6Okay, so the dot product of a and b is -6. Now, to find the angle between them, we use the formula:a · b = |a||b|cosθWe know a · b is -6, |a| is 4, |b| is 3, so:-6 = 4 * 3 * cosθSimplify:-6 = 12 cosθDivide both sides by 12:cosθ = -6 / 12 = -0.5So, cosθ = -0.5. Now, what angle has a cosine of -0.5? I remember that cos(120°) = -0.5, since 120 degrees is in the second quadrant where cosine is negative. Alternatively, in radians, that's 2π/3.Let me confirm: cos(2π/3) is indeed -0.5. Yes, that's correct.So, the angle between vectors a and b is 120 degrees or 2π/3 radians.Wait, let me just recap to make sure I didn't make any mistakes. Expanded the dot product correctly, substituted the magnitudes, solved for a · b, then used the dot product formula to find the angle. Everything seems to check out. So, the angle is 2π/3.**Final Answer**(1) boxed{-44}(2) boxed{dfrac{2pi}{3}}

answer：Okay, so I have this algebra problem to simplify the expression: 3y + 4y² - 2 - (8 - 3y - 4y²). Hmm, let me think about how to approach this step by step. First, I remember that when there's a negative sign before parentheses, it means I need to distribute that negative sign to each term inside the parentheses. So, let me rewrite the expression without the parentheses by applying the negative sign.Starting with the original expression: 3y + 4y² - 2 - (8 - 3y - 4y²).I'll distribute the negative sign to each term inside the parentheses:- The first term inside the parentheses is 8, so with the negative sign, it becomes -8.- The second term is -3y, and applying the negative sign would make it +3y.- The third term is -4y², and applying the negative sign would make it +4y².So after distributing the negative sign, the expression becomes: 3y + 4y² - 2 - 8 + 3y + 4y².Now, I need to combine like terms. Like terms are terms that have the same variable raised to the same power. Let's identify them:- The terms with y² are 4y² and +4y².- The terms with y are 3y and +3y.- The constant terms are -2 and -8.Let's combine them one by one.Starting with the y² terms: 4y² + 4y². That adds up to 8y².Next, the y terms: 3y + 3y. That adds up to 6y.Finally, the constant terms: -2 - 8. That adds up to -10.Putting it all together, the simplified expression is 8y² + 6y - 10.Wait, let me double-check my steps to make sure I didn't make any mistakes. 1. Original expression: 3y + 4y² - 2 - (8 - 3y - 4y²).2. Distribute the negative sign: 3y + 4y² - 2 - 8 + 3y + 4y².3. Combine like terms: - y² terms: 4y² + 4y² = 8y². - y terms: 3y + 3y = 6y. - Constants: -2 - 8 = -10.4. Result: 8y² + 6y - 10.Everything seems to add up correctly. I don't see any errors in my calculations. So, the simplified form of the given expression is indeed 8y² + 6y - 10.

answer：To determine the number of ways to choose a team leader and an assistant of the same gender from a class of 15 students (8 boys and 7 girls), we can break down the problem into two separate cases: selecting two boys or selecting two girls.**Case 1: Selecting Two Boys**1. **Choosing the Team Leader**: There are 8 boys to choose from for the team leader position.2. **Choosing the Assistant**: After selecting the team leader, there are 7 remaining boys to choose from for the assistant position.The number of ways to choose a team leader and an assistant from the boys is:[8 times 7 = 56]**Case 2: Selecting Two Girls**1. **Choosing the Team Leader**: There are 7 girls to choose from for the team leader position.2. **Choosing the Assistant**: After selecting the team leader, there are 6 remaining girls to choose from for the assistant position.The number of ways to choose a team leader and an assistant from the girls is:[7 times 6 = 42]**Total Number of Ways**To find the total number of ways to choose a team leader and an assistant of the same gender, we add the number of ways from both cases:[56 + 42 = 98]

answer：Okay, so I have this problem here: "If the lines ( x + 2y = 3 ) and ( nx + my = 4 ) are parallel, find ( m )." Hmm, I need to find the value of ( m ) such that these two lines are parallel. Alright, let's break this down step by step.First, I remember that two lines are parallel if they have the same slope. So, my goal is to find the slopes of these two lines and set them equal to each other. That should help me find the value of ( m ).Let's start with the first line: ( x + 2y = 3 ). I need to rewrite this in slope-intercept form, which is ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept. So, I'll solve for ( y ):( x + 2y = 3 )Subtract ( x ) from both sides:( 2y = -x + 3 )Now, divide both sides by 2:( y = (-1/2)x + 3/2 )Okay, so the slope of the first line is ( -1/2 ).Now, let's do the same for the second line: ( nx + my = 4 ). Again, I'll solve for ( y ) to get it into slope-intercept form.Starting with:( nx + my = 4 )Subtract ( nx ) from both sides:( my = -nx + 4 )Now, divide both sides by ( m ):( y = (-n/m)x + 4/m )So, the slope of the second line is ( -n/m ).Since the lines are parallel, their slopes must be equal. That means:( -1/2 = -n/m )Hmm, okay, so I can set these two slopes equal to each other:( -1/2 = -n/m )I can simplify this equation by multiplying both sides by ( -1 ) to eliminate the negative signs:( 1/2 = n/m )Now, I have:( 1/2 = n/m )I need to solve for ( m ). To do that, I can cross-multiply:( 1 * m = 2 * n )Which simplifies to:( m = 2n )Wait a second, the problem asks for ( m ), but I still have ( n ) in the equation. That means I need another equation or some additional information to find the exact value of ( m ). But looking back at the problem, I only have the two lines given. There's no other information provided about ( n ) or ( m ).Hmm, maybe I missed something. Let me double-check my steps.1. I converted both lines to slope-intercept form correctly.2. I set the slopes equal because the lines are parallel.3. I simplified the equation and ended up with ( m = 2n ).It seems like I did everything correctly, but I still can't find a numerical value for ( m ) without knowing ( n ). Maybe the problem expects me to express ( m ) in terms of ( n ), which would be ( m = 2n ). But the problem specifically asks for ( m ), not in terms of ( n ).Wait, perhaps there's a typo or missing information in the problem. If the lines are parallel, and I've correctly set their slopes equal, then ( m ) must be twice ( n ). Without additional information, I can't determine the exact value of ( m ). Maybe I need to assume a value for ( n ), but that's not ideal.Alternatively, maybe I misinterpreted the problem. Let me read it again: "If the lines ( x + 2y = 3 ) and ( nx + my = 4 ) are parallel, find ( m )." It doesn't mention anything about ( n ), so perhaps ( n ) is a known value, or maybe it's supposed to be determined as well.Wait, if I look back at the initial problem, it's presented as a question to find ( m ). Maybe there was a previous part or some context that I'm missing which gives the value of ( n ). Without that, I can't find a numerical value for ( m ).Alternatively, perhaps the problem is expecting me to express ( m ) in terms of ( n ), which I've already done: ( m = 2n ). But the problem asks to "find ( m )", which implies a numerical answer.Hmm, I'm a bit stuck here. Let me try a different approach. Maybe I can use the concept of parallel lines in terms of their coefficients.For two lines to be parallel, the coefficients of ( x ) and ( y ) must be proportional. That is, if the first line is ( a_1x + b_1y = c_1 ) and the second line is ( a_2x + b_2y = c_2 ), then for them to be parallel, ( a_1/a_2 = b_1/b_2 ).Applying this to our lines:First line: ( 1x + 2y = 3 )Second line: ( nx + my = 4 )So, the ratios of the coefficients should be equal:( 1/n = 2/m )Cross-multiplying gives:( 1 * m = 2 * n )Which simplifies to:( m = 2n )Again, I end up with the same equation: ( m = 2n ). So, unless I have a specific value for ( n ), I can't find a numerical value for ( m ).Wait, maybe the problem assumes that ( n ) is a specific value, like 1, but that's just a guess. If ( n = 1 ), then ( m = 2 ). But that's not necessarily correct unless specified.Alternatively, perhaps there's a mistake in the problem statement, and it should provide more information or specify the value of ( n ).Given all this, I think the most accurate answer I can provide is that ( m = 2n ). However, since the problem asks for ( m ), and not in terms of ( n ), I might need to reconsider.Wait, maybe I made a mistake in calculating the slopes. Let me double-check.First line: ( x + 2y = 3 )Solving for ( y ):( 2y = -x + 3 )( y = (-1/2)x + 3/2 )Slope is ( -1/2 ). That seems correct.Second line: ( nx + my = 4 )Solving for ( y ):( my = -nx + 4 )( y = (-n/m)x + 4/m )Slope is ( -n/m ). That also seems correct.Setting them equal:( -1/2 = -n/m )Simplify:( 1/2 = n/m )Cross-multiplying:( m = 2n )Yes, that's consistent.So, unless there's additional information, I can't find a numerical value for ( m ). Therefore, the answer is ( m = 2n ).But the problem asks to "find ( m )", which suggests a numerical answer. Maybe I need to assume ( n ) is given or there's a standard value. Alternatively, perhaps the problem was cut off or there's a typo.In conclusion, based on the information provided, ( m ) is twice ( n ). Without knowing ( n ), I can't provide a specific numerical value for ( m ).