💡Alright, so I have this trigonometry problem here. It says:If (2 cos α + sin α) / (cos α - 2 sin α) = -1, then tan 2α = ( )And the options are:A: 4/3B: -4/3C: -3/4D: 3/4Okay, so I need to find tan 2α given that equation. Hmm, let's see. I remember that tan 2α can be expressed in terms of tan α, so maybe I can find tan α first and then use the double angle formula.The given equation is a fraction equal to -1. Maybe I can cross-multiply to get rid of the fraction. Let me try that.So, (2 cos α + sin α) = -1 * (cos α - 2 sin α)Let me write that out:2 cos α + sin α = -cos α + 2 sin αHmm, okay, now I can bring all the terms to one side to see if I can simplify.2 cos α + sin α + cos α - 2 sin α = 0Wait, that doesn't seem right. Let me check my steps again.Wait, no, I think I made a mistake in moving the terms. Let me try again.Starting from:2 cos α + sin α = -cos α + 2 sin αI can add cos α to both sides and subtract 2 sin α from both sides to get:2 cos α + sin α + cos α - 2 sin α = 0Wait, that's the same as before. Let me combine like terms.2 cos α + cos α = 3 cos αsin α - 2 sin α = -sin αSo, 3 cos α - sin α = 0Hmm, okay, so 3 cos α = sin αIf I divide both sides by cos α, assuming cos α ≠ 0, I get:3 = tan αSo, tan α = 3Okay, that seems straightforward. Now, I need to find tan 2α. I remember the double angle formula for tangent is:tan 2α = 2 tan α / (1 - tan² α)Let me plug in tan α = 3 into this formula.tan 2α = 2*3 / (1 - 3²) = 6 / (1 - 9) = 6 / (-8) = -6/8 = -3/4So, tan 2α = -3/4Looking back at the options, that's option C.Wait, but let me double-check my steps to make sure I didn't make any mistakes.Starting with the equation:(2 cos α + sin α) / (cos α - 2 sin α) = -1Cross-multiplying:2 cos α + sin α = -cos α + 2 sin αBringing all terms to the left:2 cos α + sin α + cos α - 2 sin α = 0Combining like terms:3 cos α - sin α = 0So, 3 cos α = sin αDividing both sides by cos α:3 = tan αSo, tan α = 3Then, using the double angle formula:tan 2α = 2*3 / (1 - 9) = 6 / (-8) = -3/4Yes, that seems correct. So, the answer should be C: -3/4.I don't see any mistakes in my calculations, so I think that's the right answer.

💡Okay, so I need to calculate ((2 + 2i)^{18} - (2 - 2i)^{18}). Hmm, that seems a bit intimidating at first because of the high exponent, but maybe I can break it down step by step.First, I remember that when dealing with complex numbers raised to powers, it's often helpful to express them in polar form. Polar form uses the modulus and argument of the complex number, which can make exponentiation easier using De Moivre's theorem. So, let me try converting (2 + 2i) and (2 - 2i) into polar form.Starting with (2 + 2i):- The modulus (r) is calculated as (sqrt{a^2 + b^2}), where (a) and (b) are the real and imaginary parts, respectively. So, (r = sqrt{2^2 + 2^2} = sqrt{4 + 4} = sqrt{8} = 2sqrt{2}).- The argument (theta) is the angle made with the positive real axis. Since both the real and imaginary parts are positive, the complex number is in the first quadrant. So, (theta = arctan{left(frac{2}{2}right)} = arctan{1} = frac{pi}{4}).Therefore, (2 + 2i) in polar form is (2sqrt{2} left(cos{frac{pi}{4}} + isin{frac{pi}{4}}right)).Similarly, for (2 - 2i):- The modulus is the same, (2sqrt{2}), because squaring and adding the real and imaginary parts still gives 8.- The argument is different because the imaginary part is negative. So, it's in the fourth quadrant. The reference angle is still (frac{pi}{4}), but since it's in the fourth quadrant, the argument is (-frac{pi}{4}) or equivalently ( frac{7pi}{4} ).So, (2 - 2i) in polar form is (2sqrt{2} left(cos{left(-frac{pi}{4}right)} + isin{left(-frac{pi}{4}right)}right)) or (2sqrt{2} left(cos{frac{7pi}{4}} + isin{frac{7pi}{4}}right)).Now, I need to raise each of these to the 18th power. Using De Moivre's theorem, which states that ([r(cos{theta} + isin{theta})]^n = r^n (cos{ntheta} + isin{ntheta})), I can compute each term.Starting with ((2 + 2i)^{18}):- The modulus raised to the 18th power is ((2sqrt{2})^{18}).- The argument multiplied by 18 is (18 times frac{pi}{4} = frac{9pi}{2}).Similarly, for ((2 - 2i)^{18}):- The modulus is the same, so ((2sqrt{2})^{18}).- The argument multiplied by 18 is (18 times left(-frac{pi}{4}right) = -frac{9pi}{2}).Wait, (frac{9pi}{2}) is more than (2pi), so I should probably subtract multiples of (2pi) to find an equivalent angle between 0 and (2pi). Let's see, (frac{9pi}{2} = 4pi + frac{pi}{2}). Since (4pi) is two full rotations, the angle simplifies to (frac{pi}{2}). Similarly, (-frac{9pi}{2}) can be converted by adding multiples of (2pi). Let's add (5 times 2pi = 10pi) to (-frac{9pi}{2}), which gives (-frac{9pi}{2} + 10pi = frac{11pi}{2}). But (frac{11pi}{2}) is still more than (2pi), so subtract (2pi) again: (frac{11pi}{2} - 2pi = frac{7pi}{2}). Hmm, that's still more than (2pi). Wait, maybe I should have added (4pi) instead? Let me check: (-frac{9pi}{2} + 4pi = -frac{9pi}{2} + frac{8pi}{2} = -frac{pi}{2}). That's within the range, so the angle is (-frac{pi}{2}), which is equivalent to (frac{3pi}{2}).So, after simplifying the angles:- ((2 + 2i)^{18} = (2sqrt{2})^{18} left(cos{frac{pi}{2}} + isin{frac{pi}{2}}right))- ((2 - 2i)^{18} = (2sqrt{2})^{18} left(cos{frac{3pi}{2}} + isin{frac{3pi}{2}}right))Now, let's compute ((2sqrt{2})^{18}). First, note that (2sqrt{2} = 2^{1} times 2^{1/2} = 2^{3/2}). Therefore, ((2sqrt{2})^{18} = (2^{3/2})^{18} = 2^{(3/2) times 18} = 2^{27}). Calculating (2^{10} = 1024), (2^{20} = 1,048,576), so (2^{27} = 2^{20} times 2^7 = 1,048,576 times 128 = 134,217,728). Wait, but I think I made a mistake here. Let me double-check: (2^{10} = 1024), (2^{20} = 1,048,576), (2^{30} = 1,073,741,824). So, (2^{27} = 2^{30} / 2^3 = 1,073,741,824 / 8 = 134,217,728). Yes, that's correct.So, both terms have a modulus of (134,217,728). Now, let's compute the trigonometric parts.For ((2 + 2i)^{18}):- (cos{frac{pi}{2}} = 0)- (sin{frac{pi}{2}} = 1)So, ((2 + 2i)^{18} = 134,217,728 (0 + i times 1) = 134,217,728i).For ((2 - 2i)^{18}):- (cos{frac{3pi}{2}} = 0)- (sin{frac{3pi}{2}} = -1)So, ((2 - 2i)^{18} = 134,217,728 (0 + i times (-1)) = -134,217,728i).Now, subtracting these two:[(2 + 2i)^{18} - (2 - 2i)^{18} = 134,217,728i - (-134,217,728i) = 134,217,728i + 134,217,728i = 268,435,456i]Wait a minute, looking back at the options, option C is (268435456i), which matches my result. But I want to make sure I didn't make any mistakes in my calculations, especially with the angles and the modulus.Let me verify the modulus again. (2sqrt{2}) raised to the 18th power: ((2sqrt{2})^{18} = (2^{3/2})^{18} = 2^{(3/2) times 18} = 2^{27}). Yes, that's correct. And (2^{27}) is indeed 134,217,728.For the angles, I simplified (frac{9pi}{2}) to (frac{pi}{2}) by subtracting (4pi), which is correct because angles are periodic with period (2pi). Similarly, for (-frac{9pi}{2}), adding (4pi) (which is (8pi/2)) gives (-frac{9pi}{2} + frac{8pi}{2} = -frac{pi}{2}), which is equivalent to (frac{3pi}{2}) when considering the unit circle. So, that part is also correct.Therefore, the calculations seem accurate, and the result is (268,435,456i), which corresponds to option C.

💡Okay, so I have this problem where I need to find the cosine of the angle between vectors **b** and **c** when the magnitude of **c** is at its minimum value. Hmm, let me try to break this down step by step.First, I remember that the cosine of the angle between two vectors can be found using the dot product formula. Specifically, cosθ = ( **b** · **c** ) / ( |**b**| |**c**| ). So, I need to figure out **c** when its magnitude is minimized and then compute this dot product.But wait, the problem mentions that **c** is expressed in terms of **a** and **b**. It says **c** = x**a** + (1 - x)**b**. Okay, so **c** is a linear combination of **a** and **b** with a parameter x. My goal is to find the value of x that minimizes |**c**|, and then use that to find the cosine of the angle between **b** and **c**.Alright, let's start by recalling that the magnitude squared of a vector **v** is **v** · **v**. So, |**c**|² = (**c**) · (**c**). Let's write that out:|**c**|² = (x**a** + (1 - x)**b**) · (x**a** + (1 - x)**b**)Expanding this, we get:|**c**|² = x²(**a** · **a**) + 2x(1 - x)(**a** · **b**) + (1 - x)²(**b** · **b**)I need to compute each of these dot products. Let me note down the given vectors:**a** = (3, -1), so **a** · **a** = 3² + (-1)² = 9 + 1 = 10.**b** is given as (1, 2), so **b** · **b** = 1² + 2² = 1 + 4 = 5.Now, **a** · **b** = (3)(1) + (-1)(2) = 3 - 2 = 1.Wait, hold on, the initial problem didn't specify **b**, but in the previous thought process, **b** was (1, 2). I think I need to confirm if **b** is indeed (1, 2). Let me check the problem statement again.Oh, wait, actually, in the initial problem, **b** is given as (1, 2). So, okay, that's correct. So, **a** · **b** = 1.So, plugging these values back into the expression for |**c**|²:|**c**|² = x²(10) + 2x(1 - x)(1) + (1 - x)²(5)Simplify each term:First term: 10x²Second term: 2x(1 - x) = 2x - 2x²Third term: 5(1 - 2x + x²) = 5 - 10x + 5x²Now, combine all terms:10x² + (2x - 2x²) + (5 - 10x + 5x²)Combine like terms:10x² - 2x² + 5x² = 13x²2x - 10x = -8xAnd the constant term is 5.So, |**c**|² = 13x² - 8x + 5.Wait, that doesn't seem right because in the initial thought process, it was 25x² - 20x + 5. Did I make a mistake?Let me double-check the expansion:|**c**|² = x²(**a** · **a**) + 2x(1 - x)(**a** · **b**) + (1 - x)²(**b** · **b**)So, plugging in:x²(10) + 2x(1 - x)(1) + (1 - x)²(5)So, 10x² + 2x - 2x² + 5(1 - 2x + x²)Which is 10x² + 2x - 2x² + 5 - 10x + 5x²Combine like terms:10x² - 2x² + 5x² = 13x²2x - 10x = -8xConstant term: 5So, yes, |**c**|² = 13x² - 8x + 5.But in the initial thought process, it was 25x² - 20x + 5. Hmm, that suggests that maybe **a** · **b** was different. Wait, in the initial problem, was **a** · **b** equal to -5?Wait, let me check the initial problem again. The user wrote:"Given **a** = (3, -1), if **a** ⊥ **c**, then (**a** · **c**) = 0.So, (**a** · (x**a** + (1 - x)**b**)) = 0Which is x(**a** · **a**) + (1 - x)(**a** · **b**) = 0So, 10x + (1 - x)(-5) = 0Thus, x = 1/3."Wait, so in the initial problem, **a** · **b** was -5, but in my calculation, **a** · **b** is 1. That's conflicting.Wait, hold on, maybe **b** is different. In the initial problem, was **b** given as (1, 2) or something else? Let me check.Looking back, the user wrote:"Given **a** = (3, -1), if **a** ⊥ **c**, then (**a** · **c**) = 0.So, (**a** · (x**a** + (1 - x)**b**)) = 0Which is x(**a** · **a**) + (1 - x)(**a** · **b**) = 0So, 10x + (1 - x)(-5) = 0Thus, x = 1/3."So, from this, **a** · **b** = -5. But when I calculated **a** · **b** with **a** = (3, -1) and **b** = (1, 2), I got 3*1 + (-1)*2 = 3 - 2 = 1. So, that's conflicting.Wait, so either **b** is different or the initial calculation was wrong. Let me see.If **a** · **b** = -5, then given **a** = (3, -1), let's find **b**.Let **b** = (p, q). Then **a** · **b** = 3p - q = -5.So, 3p - q = -5. So, q = 3p + 5.But without more information, we can't determine **b** uniquely. So, perhaps in the initial problem, **b** was given as (1, 2), but in the thought process, **a** · **b** was -5, which would require **b** to be something else.Wait, maybe I need to clarify. Let me assume that in the initial problem, **a** · **b** = -5, so **b** is such that 3p - q = -5.But for the purpose of solving the problem, maybe I can proceed with the given information.Wait, perhaps the initial problem had **b** = (1, 2), but in the thought process, **a** · **b** was -5, which is inconsistent. So, maybe I need to adjust.Alternatively, perhaps the initial problem had **b** = (1, 2), so **a** · **b** = 1, as I calculated earlier. Then, in the thought process, the user had **a** · **b** = -5, which is conflicting.Wait, maybe the initial problem was different. Let me see.Wait, the user wrote:"Given **a** = (3, -1), if **a** ⊥ **c**, then (**a** · **c**) = 0.So, (**a** · (x**a** + (1 - x)**b**)) = 0Which is x(**a** · **a**) + (1 - x)(**a** · **b**) = 0So, 10x + (1 - x)(-5) = 0Thus, x = 1/3."So, in this case, **a** · **b** = -5.But if **a** = (3, -1), then **a** · **b** = 3p - q = -5.So, for **b** = (p, q), 3p - q = -5.So, **b** is not uniquely determined, but let's suppose that **b** is such that **a** · **b** = -5.But in the thought process, the user also wrote:"Given **a** = (3, -1), we have |**a**| = sqrt(3² + (-1)²) = sqrt(10),From **c** = x**a** + (1 - x)**b**, we get |**c**|² = (x**a** + (1 - x)**b**)²|**c**|² = x²**a** · **a** + (1 - x)²**b** · **b** + 2**a** · **b**x(1 - x) = 10x² + 5(1 - x)² - 10x(1 - x)Thus, |**c**|² = 25x² - 20x + 5 = 25(x - 2/5)² + 1Therefore, when x = 2/5, it reaches the minimum value 1."Wait, so in this case, **b** · **b** = 5, so |**b**| = sqrt(5). So, **b** is a vector with magnitude sqrt(5). And **a** · **b** = -5.So, given that, let's proceed.So, **a** = (3, -1), |**a**| = sqrt(10)**b** is a vector such that **a** · **b** = -5 and |**b**| = sqrt(5). So, **b** is a vector with these properties.So, let's define **b** as (p, q). Then:3p - q = -5 (from **a** · **b** = -5)and p² + q² = 5 (from |**b**|² = 5)So, we can solve for p and q.From the first equation: q = 3p + 5Substitute into the second equation:p² + (3p + 5)² = 5Expand:p² + 9p² + 30p + 25 = 5Combine like terms:10p² + 30p + 25 - 5 = 010p² + 30p + 20 = 0Divide by 10:p² + 3p + 2 = 0Factor:(p + 1)(p + 2) = 0So, p = -1 or p = -2Thus, if p = -1, then q = 3*(-1) + 5 = -3 + 5 = 2If p = -2, then q = 3*(-2) + 5 = -6 + 5 = -1So, **b** can be either (-1, 2) or (-2, -1)Wait, but in the initial thought process, **b** was (1, 2). Hmm, that's conflicting.Wait, perhaps in the initial problem, **b** was (1, 2), but in this case, **a** · **b** would be 1, not -5. So, perhaps the initial problem had a different **b**.Alternatively, maybe the initial problem had **b** = (-1, 2), which would give **a** · **b** = 3*(-1) + (-1)*2 = -3 - 2 = -5, which matches.So, perhaps **b** = (-1, 2). Let me confirm.If **b** = (-1, 2), then **a** · **b** = 3*(-1) + (-1)*2 = -3 - 2 = -5, which matches.And |**b**| = sqrt((-1)^2 + 2^2) = sqrt(1 + 4) = sqrt(5), which also matches.So, **b** = (-1, 2)Okay, so now, with **a** = (3, -1) and **b** = (-1, 2), let's proceed.So, **c** = x**a** + (1 - x)**b**So, **c** = x(3, -1) + (1 - x)(-1, 2)Compute **c**:**c** = (3x, -x) + (-1 + x, 2 - 2x)So, **c** = (3x -1 + x, -x + 2 - 2x) = (4x -1, -3x + 2)Wait, let me double-check:First component: 3x + (-1)(1 - x) = 3x - (1 - x) = 3x -1 + x = 4x -1Second component: -1*x + 2*(1 - x) = -x + 2 - 2x = -3x + 2Yes, so **c** = (4x -1, -3x + 2)Now, to find |**c**|², we compute (4x -1)^2 + (-3x + 2)^2Let's expand:(16x² - 8x + 1) + (9x² - 12x + 4) = 16x² + 9x² -8x -12x +1 +4 = 25x² -20x +5So, |**c**|² = 25x² -20x +5To find the minimum value, we can complete the square or take the derivative.Completing the square:25x² -20x +5 = 25(x² - (4/5)x) +5Complete the square inside the parentheses:x² - (4/5)x = x² - (4/5)x + (4/10)^2 - (4/10)^2 = (x - 2/5)^2 - (16/100)So,25(x² - (4/5)x) +5 = 25[(x - 2/5)^2 - 16/100] +5 = 25(x - 2/5)^2 - 4 +5 = 25(x - 2/5)^2 +1So, |**c**|² = 25(x - 2/5)^2 +1Thus, the minimum value occurs when x = 2/5, and the minimum |**c**|² is 1, so |**c**| =1.Okay, so when x = 2/5, |**c**| is minimized.Now, we need to find the cosine of the angle between **b** and **c** at this x.So, cosθ = (**b** · **c**) / (|**b**| |**c**|)We already know |**b**| = sqrt(5), |**c**| =1.So, we need to compute **b** · **c** when x = 2/5.First, let's find **c** when x = 2/5.From earlier, **c** = (4x -1, -3x +2)So, plug in x = 2/5:First component: 4*(2/5) -1 = 8/5 -5/5 = 3/5Second component: -3*(2/5) +2 = -6/5 +10/5 = 4/5So, **c** = (3/5, 4/5)Now, **b** = (-1, 2)Compute **b** · **c**:(-1)*(3/5) + 2*(4/5) = -3/5 + 8/5 = 5/5 =1So, **b** · **c** =1Thus, cosθ = 1 / (sqrt(5)*1) = 1/sqrt(5) = sqrt(5)/5Therefore, the cosine of the angle between **b** and **c** when |**c**| is minimized is sqrt(5)/5.Wait, let me double-check the calculations to make sure I didn't make any mistakes.First, **c** when x=2/5:4*(2/5) -1 = 8/5 -5/5 = 3/5-3*(2/5) +2 = -6/5 +10/5 =4/5So, **c**=(3/5,4/5). Correct.**b**=(-1,2)Dot product: (-1)*(3/5) +2*(4/5)= -3/5 +8/5=5/5=1. Correct.|**b**|=sqrt(5), |**c**|=1.Thus, cosθ=1/(sqrt(5)*1)=1/sqrt(5)=sqrt(5)/5. Correct.So, the final answer is sqrt(5)/5.

💡Okay, so I have this problem where John is guessing randomly on a 20-question multiple-choice test. Each question has three choices: one correct and two incorrect. I need to find the probability that he gets exactly half of the questions correct, which is 10 out of 20. Hmm, let me think about how to approach this.First, I remember that when dealing with probabilities of getting a certain number of successes in a series of independent trials, the binomial distribution is useful. Each question John answers is like a trial where success is getting the question right, and failure is getting it wrong. Since he's guessing randomly, each trial is independent, and the probability of success remains constant.So, the formula for the binomial probability is:[ P(X = k) = binom{n}{k} cdot p^k cdot (1-p)^{n-k} ]Where:- ( n ) is the total number of trials (questions), which is 20.- ( k ) is the number of successes (correct answers), which is 10.- ( p ) is the probability of success on a single trial. Since each question has three choices, the probability of guessing correctly is ( frac{1}{3} ).Alright, plugging in the values, we get:[ P(X = 10) = binom{20}{10} cdot left(frac{1}{3}right)^{10} cdot left(frac{2}{3}right)^{10} ]Now, I need to compute each part step by step.First, let's calculate the binomial coefficient ( binom{20}{10} ). This represents the number of ways to choose 10 successes out of 20 trials. The formula for the binomial coefficient is:[ binom{n}{k} = frac{n!}{k!(n - k)!} ]So, plugging in the numbers:[ binom{20}{10} = frac{20!}{10! cdot 10!} ]Calculating factorials can get really big, so I might need to use a calculator or simplify it. But I remember that ( 20! ) is 20 factorial, which is 20 × 19 × 18 × ... × 1, and similarly for 10!.But instead of calculating the entire factorials, maybe I can simplify the expression:[ frac{20!}{10! cdot 10!} = frac{20 times 19 times 18 times 17 times 16 times 15 times 14 times 13 times 12 times 11}{10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1} ]Let me compute this step by step.First, the numerator: 20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11.Calculating this:20 × 19 = 380380 × 18 = 6,8406,840 × 17 = 116,280116,280 × 16 = 1,860,4801,860,480 × 15 = 27,907,20027,907,200 × 14 = 390,700,800390,700,800 × 13 = 5,079,110,4005,079,110,400 × 12 = 60,949,324,80060,949,324,800 × 11 = 670,442,572,800So, the numerator is 670,442,572,800.Now, the denominator: 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.Calculating this:10 × 9 = 9090 × 8 = 720720 × 7 = 5,0405,040 × 6 = 30,24030,240 × 5 = 151,200151,200 × 4 = 604,800604,800 × 3 = 1,814,4001,814,400 × 2 = 3,628,8003,628,800 × 1 = 3,628,800So, the denominator is 3,628,800.Now, dividing the numerator by the denominator:670,442,572,800 ÷ 3,628,800.Let me see. Let's divide both numerator and denominator by 100 to make it easier:670,442,572,800 ÷ 100 = 6,704,425,7283,628,800 ÷ 100 = 36,288Now, 6,704,425,728 ÷ 36,288.Hmm, this is still a bit big. Maybe I can simplify further.Let me see if 36,288 divides into 6,704,425,728 evenly.First, let me see how many times 36,288 goes into 6,704,425,728.Alternatively, maybe I can use prime factorization or something, but that might take too long.Wait, I think I remember that ( binom{20}{10} ) is 184,756. Let me check that.Yes, actually, ( binom{20}{10} = 184,756 ). So, I can just use that value instead of calculating the entire factorials.Alright, so ( binom{20}{10} = 184,756 ).Next, let's compute ( left(frac{1}{3}right)^{10} ).Calculating ( left(frac{1}{3}right)^{10} ):( frac{1}{3} ) to the power of 10 is ( frac{1}{3^{10}} ).3^10 is 59,049. So, ( left(frac{1}{3}right)^{10} = frac{1}{59,049} ).Similarly, ( left(frac{2}{3}right)^{10} ) is ( frac{2^{10}}{3^{10}} ).2^10 is 1,024, so ( left(frac{2}{3}right)^{10} = frac{1,024}{59,049} ).Now, putting it all together:[ P(X = 10) = 184,756 times frac{1}{59,049} times frac{1,024}{59,049} ]Multiplying the fractions:First, multiply ( frac{1}{59,049} times frac{1,024}{59,049} ):[ frac{1 times 1,024}{59,049 times 59,049} = frac{1,024}{3,486,784,401} ]So, now we have:[ 184,756 times frac{1,024}{3,486,784,401} ]Multiplying 184,756 by 1,024:Let me compute that.First, 184,756 × 1,000 = 184,756,000Then, 184,756 × 24 = ?184,756 × 20 = 3,695,120184,756 × 4 = 739,024Adding them together: 3,695,120 + 739,024 = 4,434,144So, total is 184,756,000 + 4,434,144 = 189,190,144Wait, let me check that again.Wait, 184,756 × 1,024 is equal to 184,756 × (1,000 + 24) = 184,756,000 + (184,756 × 24).Calculating 184,756 × 24:First, 184,756 × 20 = 3,695,120Then, 184,756 × 4 = 739,024Adding those together: 3,695,120 + 739,024 = 4,434,144So, total is 184,756,000 + 4,434,144 = 189,190,144Wait, but 184,756 × 1,024 is 184,756 × 1,024.Alternatively, 184,756 × 1,024 = 184,756 × (1,000 + 24) = 184,756,000 + 4,434,144 = 189,190,144Yes, that seems correct.So, the numerator is 189,190,144.The denominator is 3,486,784,401.So, the probability is ( frac{189,190,144}{3,486,784,401} ).Now, let me see if this fraction can be simplified.To simplify, I need to find the greatest common divisor (GCD) of the numerator and the denominator.But both numbers are quite large, so it might be time-consuming. Alternatively, I can check if both numbers are divisible by small primes.First, let's see if both are even. The numerator is 189,190,144, which is even, and the denominator is 3,486,784,401, which is odd. So, 2 is not a common divisor.Next, check divisibility by 3.For the numerator: 1+8+9+1+9+0+1+4+4 = 1+8=9, 9+9=18, 18+1=19, 19+9=28, 28+0=28, 28+1=29, 29+4=33, 33+4=37. 37 is not divisible by 3, so the numerator is not divisible by 3.Denominator: 3+4+8+6+7+8+4+4+0+1 = 3+4=7, 7+8=15, 15+6=21, 21+7=28, 28+8=36, 36+4=40, 40+4=44, 44+0=44, 44+1=45. 45 is divisible by 3, so the denominator is divisible by 3.Since the numerator is not divisible by 3, but the denominator is, 3 is not a common divisor.Next, check divisibility by 5. The numerator ends with 4, so not divisible by 5. Denominator ends with 1, so not divisible by 5.Next, 7. This might be more complicated. Let me try dividing the numerator by 7.189,190,144 ÷ 7.7 × 27,000,000 = 189,000,000Subtracting: 189,190,144 - 189,000,000 = 190,144Now, 190,144 ÷ 7.7 × 27,000 = 189,000Subtracting: 190,144 - 189,000 = 1,1441,144 ÷ 7 = 163.428... So, not a whole number. Therefore, numerator is not divisible by 7.Denominator: 3,486,784,401 ÷ 7.Let me see, 7 × 498,112,057 = 3,486,784,401? Wait, 7 × 498,112,057 = 3,486,784,401? Let me check:7 × 498,112,057 = 7 × 400,000,000 = 2,800,000,0007 × 98,112,057 = 686,784,399Adding together: 2,800,000,000 + 686,784,399 = 3,486,784,399Wait, that's 3,486,784,399, which is 2 less than the denominator. So, 3,486,784,401 - 3,486,784,399 = 2. So, the denominator is not divisible by 7.So, 7 is not a common divisor.Next, 11. Let's check divisibility by 11.For the numerator: 189,190,144The rule for 11 is alternating sum of digits.Starting from the right: 4 (position 1), 4 (position 2), 1 (position 3), 0 (position 4), 9 (position 5), 1 (position 6), 9 (position 7), 8 (position 8), 1 (position 9)Wait, actually, the standard rule is to take the sum of the digits in the odd positions and subtract the sum of the digits in the even positions.So, let's write the number: 1 8 9 1 9 0 1 4 4Positions: 1 (1), 2 (8), 3 (9), 4 (1), 5 (9), 6 (0), 7 (1), 8 (4), 9 (4)Sum of odd positions (1,3,5,7,9): 1 + 9 + 9 + 1 + 4 = 24Sum of even positions (2,4,6,8): 8 + 1 + 0 + 4 = 13Difference: 24 - 13 = 11, which is divisible by 11. So, the numerator is divisible by 11.Now, check the denominator: 3,486,784,401Digits: 3 4 8 6 7 8 4 4 0 1Positions: 1 (3), 2 (4), 3 (8), 4 (6), 5 (7), 6 (8), 7 (4), 8 (4), 9 (0), 10 (1)Sum of odd positions (1,3,5,7,9): 3 + 8 + 7 + 4 + 0 = 22Sum of even positions (2,4,6,8,10): 4 + 6 + 8 + 4 + 1 = 23Difference: 22 - 23 = -1, which is not divisible by 11. So, denominator is not divisible by 11.Therefore, 11 is not a common divisor.Next prime is 13. Let me check numerator ÷ 13.189,190,144 ÷ 13.13 × 14,553,088 = 189,190,144? Let me check:13 × 14,553,088:14,553,088 × 10 = 145,530,88014,553,088 × 3 = 43,659,264Adding together: 145,530,880 + 43,659,264 = 189,190,144Yes! So, numerator is divisible by 13, and the result is 14,553,088.Now, check if denominator is divisible by 13.3,486,784,401 ÷ 13.Let me compute 3,486,784,401 ÷ 13.13 × 268,214,184 = 3,486,784,392Subtracting: 3,486,784,401 - 3,486,784,392 = 9So, remainder 9. Therefore, denominator is not divisible by 13.So, 13 is not a common divisor.Next prime is 17.Check numerator ÷ 17.189,190,144 ÷ 17.17 × 11,128,832 = 189,190,144? Let me check:17 × 11,128,832:11,128,832 × 10 = 111,288,32011,128,832 × 7 = 77,901,824Adding together: 111,288,320 + 77,901,824 = 189,190,144Yes, so numerator is divisible by 17, giving 11,128,832.Check denominator ÷ 17.3,486,784,401 ÷ 17.17 × 205,104,964 = 3,486,784,388Subtracting: 3,486,784,401 - 3,486,784,388 = 13So, remainder 13. Not divisible by 17.So, 17 is not a common divisor.Next prime is 19.Check numerator ÷ 19.189,190,144 ÷ 19.19 × 9,957,376 = 189,190,144? Let's check:19 × 9,957,376:9,957,376 × 10 = 99,573,7609,957,376 × 9 = 89,616,384Adding together: 99,573,760 + 89,616,384 = 189,190,144Yes, so numerator is divisible by 19, resulting in 9,957,376.Denominator ÷ 19:3,486,784,401 ÷ 19.19 × 183,514,968 = 3,486,784,392Subtracting: 3,486,784,401 - 3,486,784,392 = 9Remainder 9. Not divisible by 19.So, 19 is not a common divisor.Next prime is 23.Check numerator ÷ 23.189,190,144 ÷ 23.23 × 8,225,658 = 189,190,134Subtracting: 189,190,144 - 189,190,134 = 10So, remainder 10. Not divisible by 23.Denominator ÷ 23:3,486,784,401 ÷ 23.23 × 151,603,669 = 3,486,784,387Subtracting: 3,486,784,401 - 3,486,784,387 = 14Remainder 14. Not divisible by 23.So, 23 is not a common divisor.Next prime is 29.Check numerator ÷ 29.189,190,144 ÷ 29.29 × 6,523,798 = 189,190,142Subtracting: 189,190,144 - 189,190,142 = 2Remainder 2. Not divisible by 29.Denominator ÷ 29:3,486,784,401 ÷ 29.29 × 120,233,944 = 3,486,784,376Subtracting: 3,486,784,401 - 3,486,784,376 = 25Remainder 25. Not divisible by 29.So, 29 is not a common divisor.Next prime is 31.Check numerator ÷ 31.189,190,144 ÷ 31.31 × 6,099,682 = 189,190,142Subtracting: 189,190,144 - 189,190,142 = 2Remainder 2. Not divisible by 31.Denominator ÷ 31:3,486,784,401 ÷ 31.31 × 112,477,561 = 3,486,784,391Subtracting: 3,486,784,401 - 3,486,784,391 = 10Remainder 10. Not divisible by 31.So, 31 is not a common divisor.At this point, I can see that the numerator and denominator don't share any common divisors beyond 1, at least up to 31. Given the size of the numbers, it's unlikely they have any larger common divisors without specific factors. Therefore, the fraction ( frac{189,190,144}{3,486,784,401} ) is already in its simplest form.So, the probability that John gets exactly half of the questions correct is ( frac{189,190,144}{3,486,784,401} ).Wait, but let me double-check my calculations to make sure I didn't make any errors.First, binomial coefficient: ( binom{20}{10} = 184,756 ). That seems correct.Then, ( left(frac{1}{3}right)^{10} = frac{1}{59,049} ). Correct.( left(frac{2}{3}right)^{10} = frac{1,024}{59,049} ). Correct.Multiplying all together: 184,756 × (1/59,049) × (1,024/59,049) = 184,756 × 1,024 / (59,049 × 59,049) = 189,190,144 / 3,486,784,401. That seems correct.And simplifying, since we checked up to 31 and found no common divisors, so yes, that's the simplest form.Therefore, the probability is ( frac{189,190,144}{3,486,784,401} ).

💡First, I need to understand what the arithmetic square root of √81 means. The arithmetic square root typically refers to the non-negative square root of a number.Next, I'll calculate the square root of 81. Since 9 multiplied by 9 equals 81, the square root of 81 is 9.Therefore, the arithmetic square root of √81 is 9.

💡Okay, so I need to figure out the average and median sales of Ice Doudou toys for the week. Let me start by understanding what average and median mean. The average is like the sum of all the sales divided by the number of days, right? And the median is the middle number when the sales are arranged in order.First, let me list out the sales for each day:- Monday: 35- Tuesday: 47- Wednesday: 50- Thursday: 48- Friday: 42- Saturday: 60- Sunday: 68To find the average, I need to add up all these numbers and then divide by 7 because there are 7 days.So, adding them up: 35 + 47 + 50 + 48 + 42 + 60 + 68.Let me do that step by step:35 + 47 = 8282 + 50 = 132132 + 48 = 180180 + 42 = 222222 + 60 = 282282 + 68 = 350Okay, so the total sales for the week are 350.Now, to find the average, I divide this total by the number of days, which is 7.350 ÷ 7 = 50So, the average sales per day are 50.Next, I need to find the median. The median is the middle number when the numbers are arranged in order. Since there are 7 days, the median will be the 4th number after sorting.Let me arrange the sales in ascending order:35, 42, 47, 48, 50, 60, 68So, the numbers in order are: 35, 42, 47, 48, 50, 60, 68There are 7 numbers, so the median is the 4th one.Counting to the 4th number: 35 (1), 42 (2), 47 (3), 48 (4)So, the median is 48.Wait, but looking back at the options, option C says average is 50 and median is 48, which matches what I got. But let me double-check to make sure I didn't make any mistakes.Total sales: 35 + 47 + 50 + 48 + 42 + 60 + 68 = 350Average: 350 ÷ 7 = 50Sorted sales: 35, 42, 47, 48, 50, 60, 68Median: 4th number is 48Yes, that seems correct. So, the average is 50 and the median is 48.