💡Okay, so I have this problem here: The sum of four positive integers that form an arithmetic sequence is 80. I need to find the greatest possible second term among all such possible sequences. Hmm, let me think about how to approach this.First, I remember that an arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is 'a' and the common difference is 'd', then the four terms would be: a, a + d, a + 2d, and a + 3d. That makes sense.Now, the sum of these four terms is given as 80. So, I can write an equation for that. Let me add them up:a + (a + d) + (a + 2d) + (a + 3d) = 80Simplifying that, I get:4a + 6d = 80Hmm, okay. Let me write that down:4a + 6d = 80I can simplify this equation by dividing both sides by 2 to make the numbers smaller and easier to work with:2a + 3d = 40Alright, so now I have 2a + 3d = 40. I need to find the greatest possible second term. The second term in the sequence is a + d. So, I want to maximize a + d.Hmm, so how can I express a + d in terms of this equation? Maybe I can solve for one variable in terms of the other. Let me try solving for d in terms of a.Starting with 2a + 3d = 40, I can subtract 2a from both sides:3d = 40 - 2aThen, divide both sides by 3:d = (40 - 2a)/3Okay, so d is equal to (40 - 2a)/3. Since we're dealing with positive integers, both a and d have to be positive integers. That means (40 - 2a) must be divisible by 3, and the result must be positive.So, for d to be an integer, (40 - 2a) must be divisible by 3. Let me write that condition down:(40 - 2a) ≡ 0 mod 3Which means 40 ≡ 2a mod 3Simplifying 40 mod 3: 40 divided by 3 is 13 with a remainder of 1. So, 40 ≡ 1 mod 3.Therefore, 1 ≡ 2a mod 3I can rewrite this as:2a ≡ 1 mod 3To solve for a, I need to find the multiplicative inverse of 2 mod 3. The inverse of 2 mod 3 is 2 because 2*2=4 ≡1 mod 3.So, multiplying both sides by 2:a ≡ 2*1 ≡ 2 mod 3Therefore, a must be congruent to 2 modulo 3. That means a can be written as:a = 3k + 2, where k is a non-negative integer.Since a must be a positive integer, k can be 0, 1, 2, etc., but we also need to ensure that d is positive.Let me express d in terms of k. Since a = 3k + 2, substituting into d:d = (40 - 2*(3k + 2))/3= (40 - 6k - 4)/3= (36 - 6k)/3= 12 - 2kSo, d = 12 - 2kNow, since d must be a positive integer, 12 - 2k > 0Which implies:12 > 2k6 > kSo, k must be less than 6. Since k is a non-negative integer, k can be 0, 1, 2, 3, 4, or 5.Now, let's list the possible values of a and d for each k:For k = 0:a = 3*0 + 2 = 2d = 12 - 2*0 = 12Sequence: 2, 14, 26, 38Second term: 14For k = 1:a = 3*1 + 2 = 5d = 12 - 2*1 = 10Sequence: 5, 15, 25, 35Second term: 15For k = 2:a = 3*2 + 2 = 8d = 12 - 2*2 = 8Sequence: 8, 16, 24, 32Second term: 16For k = 3:a = 3*3 + 2 = 11d = 12 - 2*3 = 6Sequence: 11, 17, 23, 29Second term: 17For k = 4:a = 3*4 + 2 = 14d = 12 - 2*4 = 4Sequence: 14, 18, 22, 26Second term: 18For k = 5:a = 3*5 + 2 = 17d = 12 - 2*5 = 2Sequence: 17, 19, 21, 23Second term: 19Okay, so as k increases from 0 to 5, the second term increases from 14 to 19. So, the maximum second term is 19 when k = 5.Wait, but let me double-check if k can be higher. Earlier, I concluded that k must be less than 6, so k = 5 is the maximum. If I try k = 6:a = 3*6 + 2 = 20d = 12 - 2*6 = 0But d has to be positive, so d = 0 is not allowed. Hence, k cannot be 6.Therefore, the maximum second term is 19.But just to make sure, let me verify the sequence when k = 5:a = 17, d = 2Sequence: 17, 19, 21, 23Sum: 17 + 19 + 21 + 23 = 80Yes, that adds up correctly. So, the second term is indeed 19, and it's the largest possible.I think that's the answer. But let me think if there's another way to approach this problem or if I missed any possible sequences.Another approach: Since the sum is 80, and there are four terms, the average is 20. In an arithmetic sequence, the average is equal to the average of the first and last term, which is also equal to the average of the second and third term.So, (a + (a + 3d))/2 = 20Which simplifies to (2a + 3d)/2 = 20Multiplying both sides by 2: 2a + 3d = 40Which is the same equation I had earlier. So, that's consistent.Therefore, I think my approach is correct, and the maximum second term is 19.**Final Answer**The greatest possible second term is boxed{19}.

💡Okay, so I have this problem where I need to figure out which of the given functions satisfy two specific properties. Let me first make sure I understand what each property means.Property 1 says that ( f(x+2) ) is an even function. Hmm, even functions have symmetry about the y-axis, right? So, if ( f(x+2) ) is even, that means ( f(x+2) = f(-x+2) ). Wait, no, actually, more accurately, an even function satisfies ( f(-x) = f(x) ). So, if ( f(x+2) ) is even, then ( f(-x+2) = f(x+2) ). That might mean that the function ( f ) is symmetric about the line ( x = 2 ). Let me think about that. If I shift the function ( f ) two units to the left, it becomes even, which is symmetric about the y-axis. So, the original function ( f ) must be symmetric about ( x = 2 ). That makes sense.Property 2 states that ( f(x) ) is decreasing on ( (-infty, 2) ) and increasing on ( (2, +infty) ). So, the function has a minimum at ( x = 2 ). It's like a V-shape or a parabola opening upwards with the vertex at ( x = 2 ).Now, let me look at the functions given:1. ( f(x) = |x + 2| )2. ( f(x) = (x - 2)^2 )3. ( f(x) = cos(x - 2) )I need to check each function against both properties.Starting with function ①: ( f(x) = |x + 2| )First, let's check Property 1: Is ( f(x+2) ) even?Compute ( f(x+2) ):( f(x+2) = |(x + 2) + 2| = |x + 4| )Is ( |x + 4| ) an even function? An even function satisfies ( f(-x) = f(x) ). Let's test it:( f(-x) = |-x + 4| = | -x + 4 | = |4 - x| )Compare this to ( f(x) = |x + 4| ). These are not the same unless ( x = 0 ). For example, if ( x = 1 ), ( f(1) = |1 + 4| = 5 ) and ( f(-1) = |-1 + 4| = 3 ). So, ( f(-1) neq f(1) ), meaning ( |x + 4| ) is not even. Therefore, function ① does not satisfy Property 1.Wait, but maybe I made a mistake. Let me think again. If ( f(x+2) ) is even, then ( f(x+2) = f(-x+2) ). So, for function ①, ( f(x+2) = |x + 4| ) and ( f(-x + 2) = | -x + 2 + 2 | = | -x + 4 | = |x - 4| ). So, ( |x + 4| ) vs. ( |x - 4| ). These are not equal unless ( x = 0 ). So, indeed, function ① does not satisfy Property 1.Now, just for thoroughness, let's check Property 2 for function ①. Is ( f(x) = |x + 2| ) decreasing on ( (-infty, 2) ) and increasing on ( (2, +infty) )?The function ( |x + 2| ) has a V-shape with the vertex at ( x = -2 ). So, it's decreasing on ( (-infty, -2) ) and increasing on ( (-2, +infty) ). But Property 2 requires it to be decreasing on ( (-infty, 2) ) and increasing on ( (2, +infty) ). Since the vertex is at ( x = -2 ), not ( x = 2 ), function ① doesn't satisfy Property 2 either. So, ① is out.Moving on to function ②: ( f(x) = (x - 2)^2 )First, check Property 1: Is ( f(x+2) ) even?Compute ( f(x+2) ):( f(x+2) = (x + 2 - 2)^2 = x^2 )Is ( x^2 ) an even function? Yes, because ( (-x)^2 = x^2 ). So, ( f(x+2) ) is indeed even. So, Property 1 is satisfied.Now, check Property 2: Is ( f(x) = (x - 2)^2 ) decreasing on ( (-infty, 2) ) and increasing on ( (2, +infty) )?The function ( (x - 2)^2 ) is a parabola opening upwards with vertex at ( x = 2 ). So, it's decreasing for ( x < 2 ) and increasing for ( x > 2 ). That exactly matches Property 2. So, function ② satisfies both properties.Now, function ③: ( f(x) = cos(x - 2) )First, check Property 1: Is ( f(x+2) ) even?Compute ( f(x+2) ):( f(x+2) = cos((x + 2) - 2) = cos(x) )Is ( cos(x) ) an even function? Yes, because ( cos(-x) = cos(x) ). So, Property 1 is satisfied.Now, check Property 2: Is ( f(x) = cos(x - 2) ) decreasing on ( (-infty, 2) ) and increasing on ( (2, +infty) )?Hmm, cosine functions are periodic and oscillate. Let me analyze the behavior around ( x = 2 ).The function ( cos(x - 2) ) is a cosine wave shifted to the right by 2 units. The standard cosine function has maxima at ( x = 0, 2pi, 4pi, ) etc., and minima at ( x = pi, 3pi, ) etc. So, after shifting, the maxima are at ( x = 2, 2 + 2pi, 2 + 4pi, ) etc., and minima at ( x = 2 + pi, 2 + 3pi, ) etc.So, around ( x = 2 ), the function ( cos(x - 2) ) is at a maximum. Let's see the behavior just before and after ( x = 2 ).To the left of ( x = 2 ) (i.e., in ( (-infty, 2) )), the function is decreasing because it's going from higher values towards the maximum at ( x = 2 ). Wait, no, actually, at ( x = 2 ), it's a maximum, so just before ( x = 2 ), the function is increasing towards the maximum. Wait, that doesn't make sense. Let me think again.Wait, no, the derivative of ( cos(x - 2) ) is ( -sin(x - 2) ). So, let's compute the derivative at points around ( x = 2 ).At ( x = 2 ), the derivative is ( -sin(0) = 0 ). To the left of ( x = 2 ), say ( x = 2 - epsilon ), the derivative is ( -sin(-epsilon) = sin(epsilon) ), which is positive. So, the function is increasing as it approaches ( x = 2 ) from the left.To the right of ( x = 2 ), say ( x = 2 + epsilon ), the derivative is ( -sin(epsilon) ), which is negative. So, the function is decreasing as it moves away from ( x = 2 ) to the right.Wait, that's the opposite of what Property 2 requires. Property 2 says the function should be decreasing on ( (-infty, 2) ) and increasing on ( (2, +infty) ). But for ( f(x) = cos(x - 2) ), it's increasing on ( (-infty, 2) ) and decreasing on ( (2, +infty) ). So, it's the opposite behavior.Therefore, function ③ does not satisfy Property 2. Even though it satisfies Property 1, it fails Property 2.So, summarizing:- Function ① fails both properties.- Function ② satisfies both properties.- Function ③ satisfies Property 1 but fails Property 2.Therefore, only function ② meets both given properties.

💡Okay, let's see. Vanessa has a business and she's been sending out orders. She uses packing peanuts to pack these orders. She has two types of orders: large and small. Each large order needs 200 grams of packing peanuts, and each small order needs 50 grams. She used a total of 800 grams of packing peanuts. She knows she sent out 3 large orders, and she wants to find out how many small orders she sent. First, I need to figure out how much packing peanuts were used for the large orders. Since each large order uses 200 grams and she sent 3 of them, I can multiply 200 grams by 3. 200 grams * 3 = 600 grams.So, 600 grams were used for the large orders. Now, to find out how much was used for the small orders, I subtract the amount used for large orders from the total amount used.800 grams (total) - 600 grams (large orders) = 200 grams.Now, I know that each small order uses 50 grams of packing peanuts. So, to find out how many small orders she sent, I can divide the remaining packing peanuts by the amount used per small order.200 grams / 50 grams per small order = 4 small orders.So, Vanessa sent out 4 small orders.

💡Okay, so I have this problem here about a rectangle with a length to width ratio of 5:2. It says that the rectangle has a diagonal of length d, and the area can be expressed as k times d squared. I need to find the value of k. Hmm, let me think about how to approach this.First, I know that in a rectangle, the diagonal forms a right triangle with the length and the width. So, if I can express the length and width in terms of some variable, I can use the Pythagorean theorem to relate them to the diagonal. Since the ratio of length to width is 5:2, I can let the length be 5x and the width be 2x, where x is a common multiplier.So, length = 5x and width = 2x. Now, the diagonal d can be found using the Pythagorean theorem:d² = (5x)² + (2x)²Let me compute that:d² = 25x² + 4x²d² = 29x²Okay, so from this equation, I can solve for x²:x² = d² / 29Now, the area of the rectangle is length multiplied by width, which is:Area = (5x)(2x) = 10x²But I have x² in terms of d², so I can substitute that in:Area = 10 * (d² / 29) = (10/29) * d²So, the area is (10/29) times d squared, which means k is 10/29.Wait, let me double-check my steps to make sure I didn't make a mistake. I set the length as 5x and width as 2x based on the ratio. Then, using the Pythagorean theorem, I found that d squared equals 29x squared, so x squared is d squared over 29. Then, the area is 10x squared, which becomes 10 times d squared over 29. That seems right.I think I did everything correctly. The key was to express both the length and width in terms of x, use the Pythagorean theorem to relate x to the diagonal, and then substitute back into the area formula. This way, the area is expressed in terms of d squared, and I can identify the constant k as 10/29.Yeah, I'm pretty confident that k is 10/29. I don't see any errors in my calculations or reasoning.**Final Answer**boxed{A}

💡To find the minimum value of the polynomial ( x^2 + x + 1 ), I will complete the square.First, I take the quadratic expression ( x^2 + x + 1 ) and rewrite it by completing the square. I add and subtract ( left(frac{1}{2}right)^2 = frac{1}{4} ) to complete the square.This transforms the expression into ( left(x + frac{1}{2}right)^2 + frac{3}{4} ).The term ( left(x + frac{1}{2}right)^2 ) is a perfect square and its minimum value is 0 when ( x = -frac{1}{2} ).Therefore, the minimum value of the polynomial ( x^2 + x + 1 ) is ( frac{3}{4} ).

💡Okay, so I have this problem about a biased coin that's flipped six times. The probability of getting exactly two heads is equal to the probability of getting exactly three heads. I need to find the probability of getting exactly four heads out of six flips, express it as a reduced fraction (frac{i}{j}), and then find (i + j).First, let me recall that for a biased coin, the probability of getting exactly (k) heads in (n) flips is given by the binomial probability formula:[P(k) = binom{n}{k} p^k (1-p)^{n-k}]where (p) is the probability of getting a head on a single flip, and (binom{n}{k}) is the binomial coefficient.In this problem, (n = 6), and we're told that (P(2) = P(3)). So, setting up the equation:[binom{6}{2} p^2 (1-p)^4 = binom{6}{3} p^3 (1-p)^3]Let me compute the binomial coefficients:- (binom{6}{2} = frac{6!}{2!4!} = 15)- (binom{6}{3} = frac{6!}{3!3!} = 20)So plugging those in:[15 p^2 (1-p)^4 = 20 p^3 (1-p)^3]Hmm, I can simplify this equation. Let me divide both sides by (p^2 (1-p)^3) assuming (p neq 0) and (p neq 1) (since otherwise, the probabilities would be trivial and the problem wouldn't make much sense).Dividing both sides:[15 (1 - p) = 20 p]Simplify further:[15 - 15p = 20p]Combine like terms:[15 = 35p]So, solving for (p):[p = frac{15}{35} = frac{3}{7}]Okay, so the probability of getting a head on a single flip is (frac{3}{7}). That makes sense because it's a biased coin, so it's not 50-50.Now, the next part is to find the probability of getting exactly four heads out of six flips. Using the same binomial formula:[P(4) = binom{6}{4} p^4 (1-p)^2]Compute the binomial coefficient:[binom{6}{4} = binom{6}{2} = 15]So,[P(4) = 15 left(frac{3}{7}right)^4 left(frac{4}{7}right)^2]Let me compute each part step by step.First, (left(frac{3}{7}right)^4):[left(frac{3}{7}right)^4 = frac{3^4}{7^4} = frac{81}{2401}]Next, (left(frac{4}{7}right)^2):[left(frac{4}{7}right)^2 = frac{16}{49}]Now, multiply these together with 15:[15 times frac{81}{2401} times frac{16}{49}]First, multiply the fractions:[frac{81}{2401} times frac{16}{49} = frac{81 times 16}{2401 times 49}]Compute the numerator:[81 times 16 = 1296]Compute the denominator:[2401 times 49]Wait, 2401 is (49^2), so 2401 x 49 is (49^3). Let me compute that:[49 times 49 = 2401][2401 times 49 = 2401 times 50 - 2401 = 120050 - 2401 = 117649]So, the denominator is 117649.Therefore, the product of the fractions is:[frac{1296}{117649}]Now, multiply by 15:[15 times frac{1296}{117649} = frac{15 times 1296}{117649}]Compute the numerator:[15 times 1296]Let me compute that step by step:1296 x 10 = 129601296 x 5 = 6480So, 12960 + 6480 = 19440So, numerator is 19440.Thus, the probability is:[frac{19440}{117649}]Now, I need to simplify this fraction to its lowest terms.First, let's see if 19440 and 117649 have any common factors.Let me factorize both numerator and denominator.Starting with the numerator, 19440.19440: Let's divide by 10 first: 19440 = 1944 x 10 = (1944) x 10Factorizing 1944:1944 ÷ 2 = 972972 ÷ 2 = 486486 ÷ 2 = 243243 ÷ 3 = 8181 ÷ 3 = 2727 ÷ 3 = 99 ÷ 3 = 3 ÷ 3 = 1So, 1944 = 2^3 x 3^5Therefore, 19440 = 2^3 x 3^5 x 2 x 5 = 2^4 x 3^5 x 5So, prime factors of numerator: 2^4 x 3^5 x 5Now, the denominator is 117649.I recall that 49^3 is 117649, so 49 is 7^2, so 49^3 is 7^6.So, 117649 = 7^6Therefore, denominator is 7^6.So, numerator is 2^4 x 3^5 x 5, denominator is 7^6.Looking at the prime factors, there are no common factors between numerator and denominator because numerator has primes 2, 3, 5 and denominator has prime 7.Therefore, the fraction is already in its simplest form.Wait, but hold on, 19440 and 117649: let me double-check if 19440 and 117649 have any common divisors.Compute GCD(19440, 117649).Since 117649 is 7^6, and 19440 is 2^4 x 3^5 x 5, which doesn't include 7, so GCD is 1.Therefore, the fraction (frac{19440}{117649}) is already in lowest terms.Wait, but in the initial problem statement, it's mentioned that the answer should be in the form (frac{i}{j}) in lowest terms, so I think that is the case here.But let me just confirm:19440 ÷ 7: 19440 ÷ 7 is approximately 2777.14, which is not an integer, so 7 is not a factor.Similarly, 117649 ÷ 2, 3, 5: none of these will divide evenly because it's a power of 7.So yes, the fraction is reduced completely.Therefore, the probability is (frac{19440}{117649}).But wait, in the initial problem, the user had a different thought process where they simplified it further to (frac{40}{243}). Hmm, that seems conflicting with my result. Let me check where I might have gone wrong.Wait, let me recast the problem.Wait, the user's thought process was:They had 15*(3/7)^4*(4/7)^2, which is 15*(81/2401)*(16/49) = 15*(1296/117649) = 19440/117649.Then, they tried to simplify it further, getting 2160/13072 and then 40/243.But wait, 19440 divided by 486 is 40, and 117649 divided by 486 is approximately 242.28, which is not an integer. So, perhaps their simplification was incorrect.Wait, let me check:19440 ÷ 486: 486 x 40 = 19440, correct.117649 ÷ 486: 486 x 242 = 117, 492, which is more than 117649. Wait, 486 x 242 = 486*(200 + 40 + 2) = 97200 + 19440 + 972 = 97200 + 19440 = 116640 + 972 = 117612. Then, 117649 - 117612 = 37. So, 486 doesn't divide 117649 evenly.Therefore, the simplification to 40/243 is incorrect because 486 is not a common divisor of both numerator and denominator.Therefore, the fraction (frac{19440}{117649}) is indeed in its simplest form.Wait, but 19440 and 117649: Let me compute GCD(19440, 117649).Using the Euclidean algorithm:GCD(117649, 19440)Compute 117649 ÷ 19440: 19440 x 6 = 116,640. Subtract: 117,649 - 116,640 = 1,009.So, GCD(19440, 1009)Now, compute 19440 ÷ 1009.1009 x 19 = 19,171. 19440 - 19,171 = 269.So, GCD(1009, 269)Compute 1009 ÷ 269: 269 x 3 = 807. Subtract: 1009 - 807 = 202.GCD(269, 202)269 ÷ 202 = 1 with remainder 67.GCD(202, 67)202 ÷ 67 = 3 with remainder 1.GCD(67, 1) = 1.Therefore, GCD is 1. So, the fraction cannot be simplified further.Therefore, the probability is (frac{19440}{117649}), and (i + j = 19440 + 117649 = 137089).Wait, but that seems quite large. Maybe I made a mistake earlier.Wait, let me double-check the initial setup.We had:[binom{6}{2} p^2 (1-p)^4 = binom{6}{3} p^3 (1-p)^3]Which simplifies to:15 p^2 (1-p)^4 = 20 p^3 (1-p)^3Divide both sides by p^2 (1-p)^3:15 (1 - p) = 20 pWhich leads to:15 - 15p = 20p15 = 35pp = 15/35 = 3/7So that's correct.Then, P(4) = (binom{6}{4} (3/7)^4 (4/7)^2)Which is 15*(81/2401)*(16/49) = 15*(1296/117649) = 19440/117649So, that's correct.Therefore, the reduced fraction is 19440/117649, and since GCD is 1, i + j = 19440 + 117649 = 137089.But wait, in the initial problem, the user had a different answer, 40/243, leading to 283. So, perhaps I made a mistake in the calculation.Wait, let me compute 19440 divided by 486: 19440 ÷ 486 = 40.117649 ÷ 486: Let me compute 486 x 242 = 117,612, as before. 117,649 - 117,612 = 37. So, 117649 = 486 x 242 + 37, so it's not divisible by 486.Therefore, 486 is not a common factor. So, perhaps the user made a mistake in their simplification.Alternatively, maybe I made a mistake in calculating the numerator and denominator.Wait, let me recompute the multiplication:15*(81/2401)*(16/49)First, 15*(81) = 1215Then, 1215*(16) = 19440Denominator: 2401*49 = 117649So, 19440/117649 is correct.Alternatively, perhaps the problem was intended to have a simpler fraction, so maybe I made a mistake in interpreting the problem.Wait, let me check the initial problem again."When a certain biased coin is flipped six times, the probability of getting heads exactly twice is equal to the probability of getting heads exactly three times. Let (frac{i}{j}), in lowest terms, be the probability that the coin comes up heads in exactly 4 out of 6 flips. Find (i + j)."So, the setup is correct.Wait, perhaps I made a mistake in calculating the binomial coefficients or exponents.Wait, (binom{6}{4}) is 15, correct.((3/7)^4 = 81/2401), correct.((4/7)^2 = 16/49), correct.Multiply all together: 15*(81/2401)*(16/49) = 15*(1296/117649) = 19440/117649.Yes, that's correct.Alternatively, perhaps the problem expects the answer in terms of the reduced fraction, but 19440 and 117649 have a common factor that I'm missing.Wait, 19440 is 19440 = 1944 * 10 = (81 * 24) * 10 = 81 * 240 = 81 * 240.Wait, 81 is 3^4, 240 is 16 * 15 = 16 * 3 * 5.So, 19440 = 3^4 * 16 * 3 * 5 = 3^5 * 16 * 5.Denominator: 117649 = 7^6.So, no common factors.Therefore, the fraction is indeed reduced.Wait, but 19440/117649: Let me see if both are divisible by 3.19440: 1+9+4+4+0 = 18, which is divisible by 3.117649: 1+1+7+6+4+9 = 28, which is not divisible by 3.Therefore, 3 is not a common factor.Similarly, 5: numerator ends with 0, denominator doesn't, so 5 is not a common factor.7: numerator doesn't end with 0 or 5, but let me check if 7 divides 19440.19440 ÷ 7: 7*2777 = 19439, so 19440 ÷ 7 = 2777.142..., so not an integer. Therefore, 7 is not a factor.Therefore, the fraction cannot be reduced further.So, the probability is 19440/117649, and i + j = 19440 + 117649 = 137089.But that seems like a very large number, and the initial problem didn't specify any constraints on the answer's size, but usually, these problems have more manageable numbers.Wait, perhaps I made a mistake in the initial setup.Wait, let me check the initial equation again.We have:[binom{6}{2} p^2 (1-p)^4 = binom{6}{3} p^3 (1-p)^3]Which simplifies to:15 p^2 (1-p)^4 = 20 p^3 (1-p)^3Divide both sides by p^2 (1-p)^3:15 (1 - p) = 20 pWhich leads to:15 - 15p = 20p15 = 35pp = 3/7That seems correct.Alternatively, perhaps the problem was about getting exactly two heads vs exactly three heads, but in the problem statement, it's exactly twice vs exactly three times, so that's correct.Alternatively, perhaps I made a mistake in calculating the probability for four heads.Wait, let me recast the problem.Alternatively, perhaps the user had a different approach.Wait, in the initial problem, the user had:15*(3/7)^4*(4/7)^2 = 15*(81/2401)*(16/49) = 15*(1296/117649) = 19440/117649Then, they tried to simplify it to 2160/13072 and then to 40/243.But 19440 ÷ 486 = 40, and 117649 ÷ 486 is not an integer, as we saw.Alternatively, perhaps they made a mistake in simplifying.Alternatively, perhaps I made a mistake in the calculation.Wait, let me compute 15*(81/2401)*(16/49):First, 81/2401 is approximately 0.0337, and 16/49 is approximately 0.3265.So, 0.0337 * 0.3265 ≈ 0.011.Then, 15 * 0.011 ≈ 0.165.So, the probability is approximately 0.165.Now, 40/243 is approximately 0.1646, which is very close to 0.165.So, perhaps the user thought that 19440/117649 simplifies to 40/243, but that's incorrect because 19440/117649 is approximately 0.165, and 40/243 is approximately 0.1646, which is very close but not exactly the same.Wait, let me compute 19440/117649:19440 ÷ 117649 ≈ 0.16540/243 ≈ 0.1646So, they are very close but not equal.Therefore, perhaps the user made a mistake in simplifying the fraction.Alternatively, perhaps the problem was intended to have p = 3/7, and then P(4) = 15*(3/7)^4*(4/7)^2.But let me compute 15*(3^4)*(4^2)/(7^6):3^4 = 81, 4^2 = 16, 7^6 = 117649.So, 15*81*16 = 15*1296 = 19440.Therefore, 19440/117649 is correct.Alternatively, perhaps the user thought that 19440/117649 reduces to 40/243, but that's incorrect.Wait, 40/243 is equal to 40/(3^5) = 40/243 ≈ 0.1646.But 19440/117649 is approximately 0.165, which is very close but not exactly the same.Wait, let me compute 19440/117649:19440 ÷ 117649 ≈ 0.165.40/243 ≈ 0.1646.So, they are very close but not equal.Therefore, perhaps the user made a mistake in their simplification.Alternatively, perhaps I made a mistake in the calculation.Wait, let me compute 15*(3/7)^4*(4/7)^2 again.Compute (3/7)^4:3^4 = 81, 7^4 = 2401, so 81/2401.Compute (4/7)^2:4^2 = 16, 7^2 = 49, so 16/49.Multiply these together: (81/2401)*(16/49) = (81*16)/(2401*49) = 1296/117649.Multiply by 15: 15*1296 = 19440.So, 19440/117649.Yes, that's correct.Therefore, the fraction is 19440/117649, which cannot be simplified further, so i + j = 19440 + 117649 = 137089.But that seems like a very large number, and perhaps the problem expects a smaller answer.Wait, perhaps I made a mistake in the initial equation.Wait, let me check the initial setup again.We have P(2) = P(3):[binom{6}{2} p^2 (1-p)^4 = binom{6}{3} p^3 (1-p)^3]Which simplifies to:15 p^2 (1-p)^4 = 20 p^3 (1-p)^3Divide both sides by p^2 (1-p)^3:15 (1 - p) = 20 pWhich leads to:15 - 15p = 20p15 = 35pp = 3/7That seems correct.Alternatively, perhaps the problem was about getting exactly two heads vs exactly three heads, but in the problem statement, it's exactly twice vs exactly three times, so that's correct.Alternatively, perhaps the problem was intended to have a different setup.Alternatively, perhaps I made a mistake in calculating the binomial coefficients.Wait, (binom{6}{4}) is 15, correct.Yes, because (binom{6}{4} = binom{6}{2} = 15).So, that's correct.Alternatively, perhaps the problem was intended to have a different value for p.Wait, let me check:If p = 3/7, then 1 - p = 4/7.So, P(4) = 15*(3/7)^4*(4/7)^2 = 15*(81/2401)*(16/49) = 15*(1296/117649) = 19440/117649.Yes, that's correct.Alternatively, perhaps the problem was intended to have p = 2/5 or something else, but according to the setup, p = 3/7.Therefore, I think my calculation is correct, and the answer is 19440/117649, leading to i + j = 137089.But that seems very large, and perhaps I made a mistake in the calculation.Wait, let me compute 19440 + 117649:19440 + 117649:19,440 + 117,649 = 137,089.Yes, that's correct.Alternatively, perhaps the problem expects the answer in a different form, but I don't see another way to simplify it.Therefore, I think the correct answer is 137089.But wait, in the initial problem, the user had a different answer, 283, which is much smaller. So, perhaps I made a mistake in interpreting the problem.Wait, let me read the problem again."When a certain biased coin is flipped six times, the probability of getting heads exactly twice is equal to the probability of getting heads exactly three times. Let (frac{i}{j}), in lowest terms, be the probability that the coin comes up heads in exactly 4 out of 6 flips. Find (i + j)."So, the setup is correct.Wait, perhaps the user made a mistake in their initial thought process, simplifying the fraction incorrectly.Alternatively, perhaps I made a mistake in the calculation.Wait, let me compute 19440/117649:19440 ÷ 117649 ≈ 0.165.40/243 ≈ 0.1646.So, they are very close but not equal.Therefore, perhaps the user thought that 19440/117649 simplifies to 40/243, but that's incorrect.Therefore, the correct answer is 19440/117649, leading to i + j = 137089.But that seems very large, so perhaps I made a mistake.Wait, let me check the initial equation again.We have:15 p^2 (1-p)^4 = 20 p^3 (1-p)^3Divide both sides by p^2 (1-p)^3:15 (1 - p) = 20 pWhich leads to:15 - 15p = 20p15 = 35pp = 3/7That's correct.Alternatively, perhaps the problem was intended to have p = 2/5, but according to the setup, p = 3/7.Therefore, I think my calculation is correct.Therefore, the answer is 137089.But wait, that seems too large, so perhaps I made a mistake in the calculation.Wait, let me compute 19440 + 117649:19,440 + 117,649.19,440 + 100,000 = 119,440119,440 + 17,649 = 137,089.Yes, that's correct.Therefore, I think the correct answer is 137,089.But perhaps the problem expects a different approach.Alternatively, perhaps the user made a mistake in their initial thought process, leading to a different answer.Therefore, I think my calculation is correct, and the answer is 137,089.But given that the initial problem had a different answer, perhaps I made a mistake.Alternatively, perhaps the problem was intended to have a different setup.But according to the problem statement, I think my approach is correct.Therefore, I will conclude that the answer is 137,089.