💡Okay, so I have this geometry problem here, and I need to figure out the measure of angle QPR. Let me try to visualize the problem first. There are two triangles, PQR and PRS, both of which are isosceles. First, triangle PQR is isosceles with PQ equal to QR. That means sides PQ and QR are the same length, so the base is PR. Since it's isosceles, the base angles should be equal. The angle at Q, which is angle PQR, is given as 50 degrees. Wait, hold on. If triangle PQR has PQ equal to QR, then the base is PR, and the two equal sides are PQ and QR. So, the base angles would be at P and R. That means angles QPR and QRP are equal. Given that the sum of angles in a triangle is 180 degrees, angle PQR is 50 degrees, so the remaining 130 degrees are split equally between angles QPR and QRP. So, each of those angles should be 65 degrees. So, angle QPR is 65 degrees. Hmm, but wait, that might not be the final answer because there's another triangle involved.Now, looking at triangle PRS, which is also isosceles with PR equal to RS. So, sides PR and RS are equal, making the base PS. The angle at R, which is angle PRS, is given as 100 degrees. Again, since it's an isosceles triangle, the base angles at P and S should be equal. The sum of angles in triangle PRS is 180 degrees, so the remaining 80 degrees are split equally between angles SPR and SRP. So, each of those angles is 40 degrees. Wait, so angle SPR is 40 degrees. But angle QPR was 65 degrees, and angle SPR is part of angle QPR? Or is it the other way around? Let me think.Point R is inside triangle PQR. So, point S must be somewhere such that PRS is a triangle with R inside PQR. Hmm, maybe S is on side PQ or something? I'm not entirely sure about the exact configuration, but perhaps I can figure it out.If angle QPR is 65 degrees, and angle SPR is 40 degrees, then maybe angle QPS is 65 degrees, and angle SPR is 40 degrees, so angle QPR would be angle QPS minus angle SPR? That would make angle QPR equal to 65 minus 40, which is 25 degrees. Wait, that seems too straightforward. Let me double-check. If triangle PQR has angles at P and R equal to 65 degrees each, and triangle PRS has angles at P and S equal to 40 degrees each, then angle QPR is actually the difference between those two angles because point S is somewhere along the extension or inside the triangle. Alternatively, maybe I should draw a diagram to visualize this better. Since I can't draw here, I'll try to imagine it. Triangle PQR with PQ=QR, angle at Q is 50 degrees, so the other two angles are 65 each. Point R is inside this triangle, and triangle PRS is also isosceles with PR=RS, angle at R is 100 degrees, so the other two angles are 40 each.So, point S is somewhere such that PR=RS, forming triangle PRS inside or connected to triangle PQR. Since R is inside PQR, S must be outside of PQR or on one of its sides. If PR=RS, then S is somewhere such that RS equals PR. Wait, maybe S is on side PQ? If that's the case, then angle SPR would be part of angle QPR. So, angle QPR is 65 degrees, and angle SPR is 40 degrees, so angle QPS would be 65 - 40 = 25 degrees. That seems to make sense.Alternatively, maybe S is on side QR? If S is on QR, then angle PRS is 100 degrees, which is inside triangle PRS. Hmm, but I'm not sure. Wait, another approach: maybe using the Law of Sines or something. Let me think about triangle PQR first. Since PQ=QR, and angle at Q is 50 degrees, the other angles are 65 each. So, sides opposite those angles would be in proportion.In triangle PQR, sides PQ=QR, so sides opposite angles at R and P are equal. Wait, no, sides PQ and QR are equal, so the angles opposite them, which are angles at R and P, are equal. So, angles at P and R are 65 degrees each.Now, in triangle PRS, sides PR=RS, so angles at P and S are equal. Angle at R is 100 degrees, so angles at P and S are (180 - 100)/2 = 40 degrees each.So, angle at P in triangle PRS is 40 degrees. But angle at P in triangle PQR is 65 degrees. So, maybe angle QPR is the difference between these two angles? That is, 65 - 40 = 25 degrees.Wait, that seems to make sense. So, angle QPR is 25 degrees. But let me make sure I'm not making a mistake here. Is angle QPR the same as angle QPS? Or is there a different configuration? If S is on side PQ, then angle QPS would be part of angle QPR. So, angle QPR is split into angle QPS and angle SPR. If angle SPR is 40 degrees, then angle QPS would be 65 - 40 = 25 degrees. Alternatively, if S is on side QR, then angle PRS is 100 degrees, which is inside triangle PRS. But since R is inside triangle PQR, S might be outside of PQR, making angle PRS an external angle. Wait, maybe I should consider the sum of angles around point R. Since R is inside triangle PQR, the angles around R should add up to 360 degrees. But I'm not sure if that's necessary here.Alternatively, maybe using the fact that triangles PQR and PRS share side PR, and since PR=RS, maybe there's some congruence or similarity. But I don't see an immediate way to apply that.Wait, another thought: if triangle PRS has PR=RS and angle PRS=100 degrees, then triangle PRS is determined. So, sides PR=RS, angle at R is 100 degrees, so the other angles are 40 each. So, angle at P is 40 degrees in triangle PRS.But in triangle PQR, angle at P is 65 degrees. So, if angle at P in triangle PRS is 40 degrees, then angle QPR in triangle PQR is 65 degrees, which is the angle at P. But how does that relate to angle SPR?Wait, maybe angle QPR is angle QPS plus angle SPR. If angle QPR is 65 degrees, and angle SPR is 40 degrees, then angle QPS would be 65 - 40 = 25 degrees. But I'm not entirely sure if that's the correct way to split the angle. Maybe I should consider the points more carefully.Let me try to sketch this mentally. Triangle PQR with PQ=QR, angle at Q is 50 degrees. So, PQR is an isosceles triangle with apex at Q. Point R is inside this triangle, and triangle PRS is another isosceles triangle with PR=RS and angle at R is 100 degrees.So, point S must be somewhere such that RS=PR, and angle PRS=100 degrees. Since R is inside PQR, S is likely outside of PQR, forming triangle PRS.Wait, if S is outside, then angle PRS=100 degrees is an external angle relative to triangle PQR. But how does that affect angle QPR?Alternatively, maybe S is on the extension of side PQ beyond Q, making triangle PRS with S outside. Then, angle PRS=100 degrees would be at point R, which is inside PQR.Hmm, this is getting a bit confusing. Maybe I should assign coordinates to the points to make it clearer.Let me place point Q at the origin (0,0). Since triangle PQR is isosceles with PQ=QR and angle at Q is 50 degrees, I can place point P somewhere and point R somewhere.Let's assume PQ=QR=1 for simplicity. Then, using the Law of Cosines, the length of PR can be found. In triangle PQR, sides PQ=QR=1, angle at Q=50 degrees, so PR² = PQ² + QR² - 2*PQ*QR*cos(angle Q) = 1 + 1 - 2*1*1*cos50° ≈ 2 - 2*0.6428 ≈ 2 - 1.2856 ≈ 0.7144. So, PR ≈ sqrt(0.7144) ≈ 0.845.Now, triangle PRS is isosceles with PR=RS≈0.845, and angle at R is 100 degrees. So, using the Law of Cosines again, PS² = PR² + RS² - 2*PR*RS*cos(angle R) = 0.7144 + 0.7144 - 2*0.845*0.845*cos100° ≈ 1.4288 - 2*0.714* (-0.1736) ≈ 1.4288 + 0.247 ≈ 1.6758. So, PS≈sqrt(1.6758)≈1.295.But I'm not sure if this helps me find angle QPR directly. Maybe I should use the Law of Sines in triangle PRS.In triangle PRS, sides PR=RS≈0.845, angle at R=100 degrees, so angles at P and S are 40 degrees each. So, angle at P in triangle PRS is 40 degrees.But in triangle PQR, angle at P is 65 degrees. So, if angle at P in triangle PRS is 40 degrees, then angle QPR in triangle PQR is 65 degrees, which is the angle at P. But how does that relate to angle SPR?Wait, maybe angle QPR is split into two angles: one from triangle PQR and one from triangle PRS. So, angle QPR is 65 degrees, and angle SPR is 40 degrees, so the remaining angle would be 65 - 40 = 25 degrees. But I'm not entirely sure if that's the correct way to subtract the angles. Maybe I should consider the sum of angles around point P.At point P, the angles from triangle PQR and triangle PRS meet. So, angle QPR is 65 degrees in triangle PQR, and angle SPR is 40 degrees in triangle PRS. If these angles are adjacent, then the total angle around point P would be 65 + 40 = 105 degrees, but that doesn't seem right because the total around a point should be 360 degrees.Wait, no, that's only the angles in the two triangles. The actual total around point P would include other angles as well, depending on the configuration. Maybe I'm overcomplicating it.Alternatively, perhaps using the fact that triangles PQR and PRS share side PR, and since PR=RS, maybe triangle PRS is a reflection or something. But I'm not sure.Wait, another approach: since triangle PRS is isosceles with PR=RS and angle at R=100 degrees, then the base angles at P and S are 40 degrees each. So, angle SPR=40 degrees.In triangle PQR, angle at P is 65 degrees. So, if angle SPR is 40 degrees, then angle QPR is 65 - 40 = 25 degrees.That seems to make sense. So, angle QPR is 25 degrees.But let me just verify this with another method. Maybe using the Law of Sines in triangle PQR and triangle PRS.In triangle PQR, sides PQ=QR=1, angle at Q=50 degrees, so angles at P and R are 65 degrees each. So, using the Law of Sines, PR/sin50 = PQ/sin65. So, PR = (sin50/sin65)*PQ ≈ (0.7660/0.9063)*1 ≈ 0.845, which matches what I found earlier.In triangle PRS, sides PR=RS≈0.845, angle at R=100 degrees, so angles at P and S are 40 degrees each. Using the Law of Sines, PS/sin100 = PR/sin40. So, PS = (sin100/sin40)*PR ≈ (0.9848/0.6428)*0.845 ≈ 1.221*0.845 ≈ 1.037.But I'm not sure how this helps me find angle QPR. Maybe I need to consider triangle PQS or something.Wait, perhaps considering triangle PQS. If I can find the lengths of PQ, QS, and PS, I can use the Law of Cosines to find angle QPR.But I don't have enough information about point S's position relative to triangle PQR. Maybe this approach isn't the best.Alternatively, maybe using the fact that angle QPR is 65 degrees in triangle PQR, and angle SPR is 40 degrees in triangle PRS, so angle QPR is the difference, 25 degrees.I think that makes sense because point S is inside or connected in such a way that angle SPR is part of angle QPR, so subtracting gives the remaining angle.So, after considering all these approaches, I think the measure of angle QPR is 25 degrees.

💡Alright, so I have this problem here where I need to find the infinite product of a sequence defined recursively. Let me try to understand what's going on here.First, the sequence is defined by ( a_0 = frac{1}{2} ) and for each ( n geq 1 ), ( a_n = 1 + (a_{n-1} - 1)^2 ). Hmm, that seems a bit complicated, but maybe if I compute the first few terms, I can spot a pattern.Let me start by calculating ( a_1 ):[a_1 = 1 + left( a_0 - 1 right)^2 = 1 + left( frac{1}{2} - 1 right)^2 = 1 + left( -frac{1}{2} right)^2 = 1 + frac{1}{4} = frac{5}{4}]Okay, so ( a_1 = frac{5}{4} ). That makes sense. Now, let's compute ( a_2 ):[a_2 = 1 + left( a_1 - 1 right)^2 = 1 + left( frac{5}{4} - 1 right)^2 = 1 + left( frac{1}{4} right)^2 = 1 + frac{1}{16} = frac{17}{16}]Alright, ( a_2 = frac{17}{16} ). I'm noticing that each term seems to be getting closer to 1, but it's still greater than 1. Let's do one more term to see the trend:[a_3 = 1 + left( a_2 - 1 right)^2 = 1 + left( frac{17}{16} - 1 right)^2 = 1 + left( frac{1}{16} right)^2 = 1 + frac{1}{256} = frac{257}{256}]So, ( a_3 = frac{257}{256} ). It looks like each term is of the form ( frac{2^{2^n} + 1}{2^{2^n}} ). Let me check if this holds for the terms I've calculated.For ( n = 0 ):[frac{2^{2^0} + 1}{2^{2^0}} = frac{2^1 + 1}{2^1} = frac{3}{2}]Wait, that's not equal to ( a_0 = frac{1}{2} ). Hmm, maybe my initial guess is off. Let me see.Looking back, ( a_0 = frac{1}{2} ), ( a_1 = frac{5}{4} ), ( a_2 = frac{17}{16} ), ( a_3 = frac{257}{256} ). I notice that the numerators are 1, 5, 17, 257, which are all one more than powers of 2: ( 2^1 + 1 = 3 ) (but 3 isn't 5), wait, maybe not. Alternatively, 5 is ( 4 + 1 ), 17 is ( 16 + 1 ), 257 is ( 256 + 1 ). So, starting from ( a_1 ), each numerator is ( 2^{2^n} + 1 ), where ( n ) starts at 1.Wait, let's see:- ( a_1 = frac{5}{4} = frac{2^{2^1} + 1}{2^{2^1}} = frac{4 + 1}{4} = frac{5}{4} )- ( a_2 = frac{17}{16} = frac{2^{2^2} + 1}{2^{2^2}} = frac{16 + 1}{16} = frac{17}{16} )- ( a_3 = frac{257}{256} = frac{2^{2^3} + 1}{2^{2^3}} = frac{256 + 1}{256} = frac{257}{256} )So, it seems that for ( n geq 1 ), ( a_n = frac{2^{2^n} + 1}{2^{2^n}} ). But ( a_0 ) is ( frac{1}{2} ), which doesn't fit this pattern. Maybe I need to adjust the formula to include ( a_0 ).Alternatively, perhaps the formula is ( a_n = frac{2^{2^n} + 1}{2^{2^n}} ) for ( n geq 0 ), but then ( a_0 ) would be ( frac{3}{2} ), which contradicts the given ( a_0 = frac{1}{2} ). So, maybe the formula starts from ( n = 1 ).In any case, I can see that each term after ( a_0 ) is of the form ( frac{2^{2^n} + 1}{2^{2^n}} ). Now, the problem asks for the infinite product ( prod_{i=0}^infty a_i ). So, I need to compute:[prod_{i=0}^infty a_i = a_0 times a_1 times a_2 times a_3 times ldots]Given that ( a_0 = frac{1}{2} ) and ( a_n = frac{2^{2^n} + 1}{2^{2^n}} ) for ( n geq 1 ), the product becomes:[frac{1}{2} times prod_{n=1}^infty frac{2^{2^n} + 1}{2^{2^n}}]This looks like a telescoping product or something that might simplify nicely. Let me write out the first few terms of the product to see if a pattern emerges.Let's denote ( P = prod_{n=1}^infty frac{2^{2^n} + 1}{2^{2^n}} ). Then, the total product is ( frac{1}{2} times P ).Expanding ( P ):[P = frac{5}{4} times frac{17}{16} times frac{257}{256} times ldots]I recall that products of this form can sometimes be related to geometric series or other known products. Let me think about the general term ( frac{2^{2^n} + 1}{2^{2^n}} ).Notice that ( 2^{2^n} + 1 ) is a Fermat number. Fermat numbers are of the form ( F_n = 2^{2^n} + 1 ), and they are known to be pairwise coprime. However, I'm not sure if that helps directly here, but it's good to note.Now, let's consider the product ( P ):[P = prod_{n=1}^infty frac{2^{2^n} + 1}{2^{2^n}} = prod_{n=1}^infty left( 1 + frac{1}{2^{2^n}} right)]This resembles an infinite product of terms each slightly greater than 1. I wonder if this product converges or diverges. Since each term is ( 1 + frac{1}{2^{2^n}} ), and the sum of ( frac{1}{2^{2^n}} ) converges (it's a geometric series with ratio ( frac{1}{4} )), the infinite product might converge.But I need to find the exact value of this product. Let me try to express it in a different form. Notice that:[prod_{n=1}^infty left( 1 + frac{1}{2^{2^n}} right) = prod_{n=1}^infty frac{2^{2^n} + 1}{2^{2^n}}]This can be rewritten as:[prod_{n=1}^infty frac{2^{2^n} + 1}{2^{2^n}} = prod_{n=1}^infty left( 1 + frac{1}{2^{2^n}} right)]I recall that there's a product formula related to binary expansions or something similar. Let me think about binary representations. Alternatively, maybe I can relate this product to a geometric series.Wait, another approach: consider the binary expansion of 1. The product might relate to the binary digits somehow. Alternatively, think about the product as a telescoping product.Let me try to write the product up to a finite term and see if I can find a pattern.Let ( P_k = prod_{n=1}^k frac{2^{2^n} + 1}{2^{2^n}} ). Then, ( P = lim_{k to infty} P_k ).Compute ( P_1 ):[P_1 = frac{5}{4}]Compute ( P_2 ):[P_2 = frac{5}{4} times frac{17}{16} = frac{85}{64}]Compute ( P_3 ):[P_3 = frac{85}{64} times frac{257}{256} = frac{21845}{16384}]Hmm, not sure I see a pattern here. Let me compute the decimal values to see if it approaches a limit.( P_1 = 1.25 )( P_2 = 1.328125 )( P_3 ≈ 1.33203125 )( P_4 = P_3 times frac{65537}{65536} ≈ 1.33203125 times 1.0000152587890625 ≈ 1.332048828125 )It seems like the product is converging to around 1.333..., which is ( frac{4}{3} ). Wait, ( frac{4}{3} ≈ 1.333333... ). So, maybe ( P = frac{4}{3} ).If that's the case, then the total product ( prod_{i=0}^infty a_i = frac{1}{2} times frac{4}{3} = frac{2}{3} ).But wait, let me verify this more carefully. I think there's a known product formula for this kind of expression.I recall that:[prod_{n=0}^infty left( 1 + x^{2^n} right) = frac{1}{1 - x} quad text{for} quad |x| < 1]This is a well-known infinite product identity. Let me check it for ( x = frac{1}{2} ).So, if ( x = frac{1}{2} ), then:[prod_{n=0}^infty left( 1 + left( frac{1}{2} right)^{2^n} right) = frac{1}{1 - frac{1}{2}} = 2]But in our case, the product starts from ( n = 1 ), not ( n = 0 ). So, let's adjust for that.The product from ( n = 1 ) to ( infty ) would be:[prod_{n=1}^infty left( 1 + left( frac{1}{2} right)^{2^n} right) = frac{prod_{n=0}^infty left( 1 + left( frac{1}{2} right)^{2^n} right)}{1 + left( frac{1}{2} right)^{2^0}} = frac{2}{1 + frac{1}{2}} = frac{2}{frac{3}{2}} = frac{4}{3}]Yes, that matches the earlier approximation. So, ( P = frac{4}{3} ).Therefore, the total product is:[prod_{i=0}^infty a_i = a_0 times P = frac{1}{2} times frac{4}{3} = frac{2}{3}]So, the infinite product converges to ( frac{2}{3} ).I think that makes sense. The key was recognizing the infinite product identity and adjusting for the starting index. It's interesting how these recursive sequences can lead to neat product formulas.

💡First, I'll define the variables for the son's age (( S )) and the father's age (( F )).From the first statement, twice the son's age plus the father's age equals 70:[ 2S + F = 70 ]From the second statement, twice the father's age plus the son's age equals 95:[ S + 2F = 95 ]I'll solve the first equation for ( F ):[ F = 70 - 2S ]Next, I'll substitute this expression for ( F ) into the second equation:[ S + 2(70 - 2S) = 95 ]Simplifying the equation:[ S + 140 - 4S = 95 ][ -3S + 140 = 95 ][ -3S = 95 - 140 ][ -3S = -45 ][ S = 15 ]Now that I have the son's age, I'll find the father's age:[ F = 70 - 2(15) ][ F = 70 - 30 ][ F = 40 ]The father's age is 40 years old.

💡Alright, so I've got this problem here: If 2 plus a equals 5 minus b, and 5 plus b equals 8 plus a, what is 2 minus a? Hmm, okay, let's see how I can tackle this.First, I need to understand what the problem is asking. There are two equations given, and I need to find the value of 2 minus a. That means I probably need to find the value of 'a' first, and then subtract it from 2 to get the answer.Let me write down the equations to make it clearer:1. 2 + a = 5 - b2. 5 + b = 8 + aSo, we have two equations with two variables, 'a' and 'b'. This looks like a system of linear equations, and I can solve it using either substitution or elimination. I think substitution might be easier here because the equations are pretty straightforward.Starting with the first equation: 2 + a = 5 - b. Maybe I can solve for one of the variables in terms of the other. Let's try solving for 'a' first.Subtracting 2 from both sides gives me:a = 5 - b - 2a = 3 - bOkay, so 'a' equals 3 minus 'b'. Now, I can substitute this expression for 'a' into the second equation to find the value of 'b'.The second equation is: 5 + b = 8 + aSince I know that a = 3 - b, I can replace 'a' with (3 - b):5 + b = 8 + (3 - b)Let's simplify the right side:8 + 3 - b = 11 - bSo now the equation becomes:5 + b = 11 - bHmm, okay, let's solve for 'b'. I'll add 'b' to both sides to get rid of the negative 'b' on the right:5 + b + b = 11 - b + b5 + 2b = 11Now, subtract 5 from both sides:2b = 11 - 52b = 6Divide both sides by 2:b = 3Alright, so 'b' equals 3. Now that I have the value of 'b', I can substitute it back into the expression I had for 'a' to find its value.Earlier, I found that a = 3 - b. Plugging in b = 3:a = 3 - 3a = 0So, 'a' equals 0. Now, the question asks for 2 minus 'a', which is:2 - a = 2 - 0 = 2Therefore, the value of 2 - a is 2.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting with the first equation, solving for 'a' in terms of 'b' seems correct. Then substituting into the second equation also looks right. Solving for 'b' gave me 3, and plugging that back into the expression for 'a' gave me 0. Finally, calculating 2 - a as 2 - 0 is indeed 2. Everything seems to add up, so I think my answer is correct.

💡To convert the decimal number 19 to binary, I will use the division by 2 method. This involves repeatedly dividing the number by 2 and recording the remainders.First, I divide 19 by 2. The quotient is 9 and the remainder is 1.Next, I divide the quotient 9 by 2. The new quotient is 4 and the remainder is 1.Then, I divide the quotient 4 by 2. The new quotient is 2 and the remainder is 0.I continue by dividing the quotient 2 by 2. The new quotient is 1 and the remainder is 0.Finally, I divide the quotient 1 by 2. The new quotient is 0 and the remainder is 1.Now, I collect all the remainders from the last division to the first. This gives me the binary representation of 19 as 10011.Therefore, the correct answer is option C: 10011₂.

💡Okay, so I have this problem here: I need to find the probability that the product of three randomly chosen distinct integers from the set ({ -7, -3, 1, 5, 8 }) is negative. Hmm, let me think about how to approach this.First, I remember that when multiplying numbers, the product is negative if there's an odd number of negative numbers in the multiplication. So, if I have one negative and two positive numbers, the product will be negative. Similarly, if I have three negative numbers, the product will also be negative. But in this set, how many negative numbers are there?Looking at the set ({ -7, -3, 1, 5, 8 }), I see that there are two negative numbers: -7 and -3. The rest are positive: 1, 5, and 8. So, there are only two negative numbers and three positive numbers.Now, since I'm choosing three numbers, and I want the product to be negative, I need to consider the cases where the number of negative numbers is odd. That means either one negative and two positives or three negatives. But wait, in this set, there are only two negative numbers, so it's impossible to choose three negative numbers. That means the only way to get a negative product is by choosing one negative and two positive numbers.Okay, so I need to calculate how many ways I can choose one negative and two positive numbers from the set. Then, I'll divide that by the total number of ways to choose any three numbers from the set to get the probability.Let me recall how combinations work. The number of ways to choose (k) elements from a set of (n) elements is given by the combination formula:[binom{n}{k} = frac{n!}{k!(n - k)!}]So, the total number of ways to choose three numbers from the five-element set is:[binom{5}{3} = frac{5!}{3!(5 - 3)!} = frac{5 times 4 times 3!}{3! times 2!} = frac{20}{2} = 10]So, there are 10 possible ways to choose any three numbers from the set.Now, for the number of favorable outcomes, which is choosing one negative and two positive numbers. There are two negative numbers, so the number of ways to choose one negative number is:[binom{2}{1} = 2]And there are three positive numbers, so the number of ways to choose two positive numbers is:[binom{3}{2} = frac{3!}{2!(3 - 2)!} = frac{6}{2 times 1} = 3]To find the total number of favorable outcomes, I multiply these two results together because for each way of choosing a negative number, there are multiple ways to choose the positive numbers:[2 times 3 = 6]So, there are 6 favorable ways to choose one negative and two positive numbers.Therefore, the probability that the product of the three chosen numbers is negative is the number of favorable outcomes divided by the total number of possible outcomes:[frac{6}{10} = frac{3}{5}]Wait, let me double-check to make sure I didn't make any mistakes. The total number of ways to choose three numbers is indeed 10. The number of ways to choose one negative is 2, and two positives is 3, so 2 times 3 is 6. That seems right.Is there another way this could happen? Well, if I had three negative numbers, the product would also be negative, but since there are only two negatives, that's not possible here. So, yes, the only way is one negative and two positives.I think that's correct. So, the probability should be (frac{3}{5}).