answer：First, I identify the given probabilities: Xavier's success probability is ( frac{1}{3} ), Yvonne's is ( frac{1}{2} ), and Zelda's is unknown, denoted as ( P(Z) ).The problem states that the probability of Xavier and Yvonne solving the problem but not Zelda is ( 0.0625 ). This can be expressed as:[P(X text{ and } Y text{ and not } Z) = P(X) times P(Y) times (1 - P(Z)) = 0.0625]Substituting the known values of ( P(X) ) and ( P(Y) ):[left( frac{1}{3} right) times left( frac{1}{2} right) times (1 - P(Z)) = 0.0625]Simplifying the left side:[frac{1}{6} times (1 - P(Z)) = 0.0625]To solve for ( (1 - P(Z)) ), multiply both sides by 6:[1 - P(Z) = 0.0625 times 6 = 0.375]Finally, solve for ( P(Z) ):[P(Z) = 1 - 0.375 = 0.625]

answer：Alright, so I have this problem here about a finite sequence {a_n} with m terms. It defines S(n) as the sum of all terms starting from the n-th term. So, S(1) is the sum of all terms, S(2) is the sum from the second term onwards, and so on up to S(m), which would just be the last term itself.The problem states that S(n) is the sum of the first n terms of an arithmetic sequence with the first term 1 and a common difference of 2. I need to find what a_n equals when 1 ≤ n < m.Okay, let's break this down. First, I need to recall what the sum of the first n terms of an arithmetic sequence is. The formula for the sum of the first n terms, S_n, of an arithmetic sequence is given by:S_n = (n/2) * [2a + (n - 1)d]where a is the first term and d is the common difference.In this case, the first term a is 1, and the common difference d is 2. So plugging those values in, the sum S(n) becomes:S(n) = (n/2) * [2*1 + (n - 1)*2] = (n/2) * [2 + 2n - 2] = (n/2) * [2n] = n * n = n²So, S(n) is equal to n squared. That simplifies things a bit.Now, S(n) is defined as the sum of all terms starting from the n-th term in the sequence {a_n}. So, S(n) = a_n + a_{n+1} + ... + a_m.But we also have S(n) = n². So, the sum from the n-th term to the m-th term is n².Wait, but how does this help me find a_n? Maybe I can express a_n in terms of S(n) and S(n+1). Let me think.If S(n) is the sum starting from the n-th term, then S(n+1) would be the sum starting from the (n+1)-th term. So, S(n) = a_n + S(n+1). Therefore, a_n = S(n) - S(n+1).Yes, that makes sense. Because if you take the sum starting at n and subtract the sum starting at n+1, you're left with just a_n.So, a_n = S(n) - S(n+1).We already know that S(n) = n², so let's plug that in:a_n = n² - (n+1)²Let me compute that:(n+1)² = n² + 2n + 1So,a_n = n² - (n² + 2n + 1) = n² - n² - 2n - 1 = -2n - 1So, a_n equals -2n - 1.Wait, let me verify this. If I take n=1, then a_1 = -2(1) -1 = -3. Then S(1) should be the sum from the first term, which is a_1 + a_2 + ... + a_m. But S(1) is supposed to be 1² = 1. So, if a_1 is -3, then the sum of the rest of the terms must be 4 to make the total sum 1. Hmm, that seems possible.Similarly, for n=2, a_2 = -2(2) -1 = -5. Then S(2) should be 2² = 4. So, a_2 + a_3 + ... + a_m = 4. If a_2 is -5, then the sum from a_3 to a_m must be 9. That seems consistent.Wait, but let me check if this holds for a small m. Let's say m=3. Then:a_1 = -3 a_2 = -5 a_3 = -7Then S(1) = a_1 + a_2 + a_3 = -3 -5 -7 = -15, which is not equal to 1²=1. That's a problem.Hmm, so my earlier reasoning must be flawed. Maybe I misunderstood the definition of S(n).Wait, the problem says S(n) is the sum of all terms starting from the n-th term. So, S(1) is the sum of all terms, S(2) is the sum from the second term, etc. But it also says that S(n) is the sum of the first n terms of an arithmetic sequence with first term 1 and common difference 2.Wait, so S(n) is equal to both the sum from the n-th term of {a_n} and the sum of the first n terms of another arithmetic sequence.So, S(n) is defined in two ways:1. S(n) = a_n + a_{n+1} + ... + a_m 2. S(n) = sum of first n terms of an arithmetic sequence with a=1, d=2, which is n².Therefore, for each n, the sum from a_n to a_m is equal to n².So, if I consider S(1) = a_1 + a_2 + ... + a_m = 1² = 1 S(2) = a_2 + a_3 + ... + a_m = 2² = 4 S(3) = a_3 + ... + a_m = 3² = 9 ... S(m) = a_m = m²Wait, but if S(1) is 1 and S(2) is 4, then a_1 = S(1) - S(2) = 1 - 4 = -3 Similarly, a_2 = S(2) - S(3) = 4 - 9 = -5 a_3 = S(3) - S(4) = 9 - 16 = -7 ... a_{m-1} = S(m-1) - S(m) = (m-1)² - m² = -2m +1 a_m = S(m) = m²Wait, but in my earlier calculation, I thought a_n = S(n) - S(n+1) = n² - (n+1)² = -2n -1. But when I plug in n=1, I get a_1 = -3, which is correct because S(1)=1 and S(2)=4, so a_1=1-4=-3. Similarly, a_2=4-9=-5, which is -2*2 -1=-5, so that works. a_3=9-16=-7, which is -2*3 -1=-7, that works too. So, in general, for n < m, a_n = -2n -1.But wait, when n=m, a_m = S(m) = m², which is different. So, the formula a_n = -2n -1 holds for n < m, but a_m is m².But the problem asks for a_n when 1 ≤ n < m, so the answer is a_n = -2n -1.But earlier, when I tested with m=3, I got S(1)=1, which is a_1 + a_2 + a_3 = -3 -5 -7 = -15, which is not equal to 1. So, that contradicts. So, what's wrong here?Wait, maybe I misunderstood the definition of S(n). The problem says S(n) is the sum of all terms starting from the n-th term. So, S(1) is the sum from the first term, which is the entire sum, S(2) is the sum from the second term, etc.But it also says that S(n) is the sum of the first n terms of an arithmetic sequence with first term 1 and common difference 2. So, S(n) is equal to both the sum from the n-th term of {a_n} and the sum of the first n terms of another arithmetic sequence.Therefore, for each n, the sum from a_n to a_m is equal to n².So, if I consider S(1) = a_1 + a_2 + ... + a_m = 1² = 1 S(2) = a_2 + a_3 + ... + a_m = 2² = 4 S(3) = a_3 + ... + a_m = 3² = 9 ... S(m) = a_m = m²Wait, but if S(1)=1 and S(2)=4, then a_1 = S(1) - S(2) = 1 - 4 = -3 Similarly, a_2 = S(2) - S(3) = 4 - 9 = -5 a_3 = S(3) - S(4) = 9 - 16 = -7 ... a_{m-1} = S(m-1) - S(m) = (m-1)² - m² = -2m +1 a_m = S(m) = m²Wait, but when I plug in m=3, a_1=-3, a_2=-5, a_3=9. Then S(1)=a1+a2+a3=-3-5+9=1, which is correct. S(2)=a2+a3=-5+9=4, correct. S(3)=a3=9, correct. So, in this case, a3=9, which is m²=3²=9.But earlier, I thought a3=-7, which was incorrect because I was using the formula a_n=-2n-1 for n=3, but that formula is only valid for n < m. So, for n=1 and 2, a_n=-2n-1, but for n=3, a3=9.So, in general, for 1 ≤ n < m, a_n = -2n -1, and a_m = m².Therefore, the answer is a_n = -2n -1 for 1 ≤ n < m.But let me double-check with m=4.If m=4, then:a1 = -2(1) -1 = -3 a2 = -2(2) -1 = -5 a3 = -2(3) -1 = -7 a4 = 4² = 16Then S(1)=a1+a2+a3+a4=-3-5-7+16=1, correct. S(2)=a2+a3+a4=-5-7+16=4, correct. S(3)=a3+a4=-7+16=9, correct. S(4)=a4=16, correct.So, it works for m=4 as well.Therefore, the formula a_n = -2n -1 holds for 1 ≤ n < m, and a_m = m².Hence, the answer is a_n = -2n -1.

answer：To determine how much change Tom received, let's break down the transaction step by step.1. **Trade-in Value for Super Nintendo:** - Tom's Super Nintendo is worth 150. - The store offers 80% of this value as trade-in credit. - Calculation: 80% of 150 = 0.80 * 150 = 120.2. **Cost of the NES:** - The original NES is on sale for 160. - After applying the trade-in credit, the amount Tom needs to pay is: - Calculation: 160 - 120 = 40.3. **Payment and Change Calculation:** - Tom gives the store 80. - He also receives a game worth 30, which effectively reduces the amount he needs to pay. - Effective payment after receiving the game: 40 - 30 = 10. - The change Tom receives is the difference between what he gave and the effective payment: - Calculation: 80 - 10 = 70.Therefore, Tom receives boxed{70} in change.

answer：Okay, so I have this problem about a parallelogram ABCD with area 15. Points P and Q are projections of A and C onto BD, and points R and S are projections of B and D onto AC. They tell me that PQ is 6 and RS is 8. I need to find the square of the longer diagonal BD, which is denoted as d, and express it in the form m + n√p, where m, n, and p are positive integers, and p is square-free. Then, I have to find m + n + p.Alright, let's start by recalling some properties of parallelograms. In a parallelogram, opposite sides are equal, and the diagonals bisect each other. Also, the area can be expressed as the product of the base and the height. Since ABCD is a parallelogram, the area is 15, so that might come into play when dealing with the projections.Projections of points onto lines... Hmm, projections are like dropping a perpendicular from a point to a line. So, P is the foot of the perpendicular from A to BD, and Q is the foot from C to BD. Similarly, R and S are the feet from B and D onto AC.Given that PQ is 6 and RS is 8, I need to relate these lengths to the diagonals BD and AC. Since BD is the longer diagonal, and we're asked about BD squared, I should probably focus on BD.Let me visualize this. In a parallelogram, the diagonals intersect at the midpoint. Let's call the midpoint O. So, BO = OD and AO = OC. Since P and Q are projections of A and C onto BD, and R and S are projections of B and D onto AC, maybe there's some symmetry or relationship between these projections.I think it might help to assign coordinates to the points to make things more concrete. Let me place the parallelogram in a coordinate system. Let me assume that point A is at the origin (0,0). Since it's a parallelogram, I can define the coordinates of the other points based on vectors.Let me denote vector AB as (a, b) and vector AD as (c, d). Then, point B would be at (a, b), point D at (c, d), and point C, which is the sum of vectors AB and AD, would be at (a + c, b + d).The area of the parallelogram is given by the magnitude of the cross product of vectors AB and AD, which is |ad - bc| = 15.Now, let's find the equations of the diagonals BD and AC.Diagonal BD connects points B (a, b) and D (c, d). The equation of BD can be parametrized as (a + t(c - a), b + t(d - b)) for t from 0 to 1.Similarly, diagonal AC connects points A (0,0) and C (a + c, b + d). The equation of AC can be parametrized as (s(a + c), s(b + d)) for s from 0 to 1.Now, projections of points onto these diagonals. Let's start with projections onto BD.Point P is the projection of A onto BD. Since A is at (0,0), the projection P can be found by finding the foot of the perpendicular from A to BD.Similarly, Q is the projection of C onto BD. So, we can find Q by projecting point C (a + c, b + d) onto BD.Similarly, R and S are projections of B and D onto AC.Given that PQ = 6 and RS = 8, these are the distances between the projections on BD and AC respectively.This seems a bit involved. Maybe there's a better way to approach this without coordinates.Alternatively, I remember that in a parallelogram, the sum of the squares of the diagonals is equal to twice the sum of the squares of the sides. But I don't know the sides here, so maybe that's not directly helpful.Wait, but since we have projections, maybe we can relate the lengths PQ and RS to the lengths of the diagonals and the angles between them.Let me think about the projections. The projection of a point onto a line is essentially the length of the segment from the point to the line along the perpendicular. So, the distance from A to BD is the length AP, and the distance from C to BD is the length CQ.But PQ is the distance between the projections P and Q on BD. Since P and Q are both on BD, PQ is a segment on BD. Similarly, RS is a segment on AC.Wait, so PQ is the distance between the two projections on BD, which are P and Q. Similarly, RS is the distance between R and S on AC.Given that, maybe PQ and RS can be related to the lengths of the diagonals and the angles between the sides and the diagonals.Let me denote θ as the angle between side AB and diagonal BD. Then, the projection of AB onto BD would be AB cos θ. Similarly, the projection of AD onto BD would be AD cos φ, where φ is the angle between AD and BD.But since ABCD is a parallelogram, the angles θ and φ are related. Maybe θ + φ = 180°, but I'm not sure.Alternatively, perhaps I can use the fact that the projections of A and C onto BD are P and Q, and the distance between P and Q is 6. Similarly, the projections of B and D onto AC are R and S, and the distance between R and S is 8.Since ABCD is a parallelogram, the projections of A and C onto BD should be symmetric with respect to the midpoint of BD. Similarly, the projections of B and D onto AC should be symmetric with respect to the midpoint of AC.Therefore, the distance between P and Q should be twice the distance from the midpoint of BD to either P or Q. Similarly, RS is twice the distance from the midpoint of AC to R or S.Wait, that might not necessarily be true unless the projections are equidistant from the midpoint. Hmm, maybe I need to think differently.Alternatively, since P and Q are projections of A and C onto BD, and since A and C are opposite vertices, their projections onto BD might be related.Similarly, R and S are projections of B and D onto AC.Given that, perhaps the distance PQ is related to the projection of vector AC onto BD, and RS is related to the projection of vector BD onto AC.Wait, that might make sense. Let me think.The projection of vector AC onto BD would be equal to the length of AC times the cosine of the angle between AC and BD. Similarly, the projection of BD onto AC would be the length of BD times the cosine of the angle between them.But PQ is the distance between the projections of A and C onto BD, which might be equal to the projection of AC onto BD. Similarly, RS is the projection of BD onto AC.Wait, that seems promising.So, if I denote the angle between AC and BD as φ, then the projection of AC onto BD is |AC| cos φ, and the projection of BD onto AC is |BD| cos φ.Given that, PQ = |AC| cos φ = 6, and RS = |BD| cos φ = 8.So, we have two equations:1. |AC| cos φ = 62. |BD| cos φ = 8Let me denote |AC| = m and |BD| = n. Then,m cos φ = 6n cos φ = 8Dividing the second equation by the first, we get:(n cos φ) / (m cos φ) = 8 / 6 => n/m = 4/3 => n = (4/3)mSo, |BD| = (4/3)|AC|Also, from the first equation, cos φ = 6/mFrom the second equation, cos φ = 8/n = 8/(4m/3) = 6/mSo, consistent.Now, in a parallelogram, the area can also be expressed in terms of the diagonals and the angle between them. The area is (1/2) * |AC| * |BD| * sin φ.Wait, is that correct? Let me recall. The area of a parallelogram can be expressed as (1/2) * d1 * d2 * sin θ, where d1 and d2 are the diagonals and θ is the angle between them.Yes, that's correct.Given that, the area is 15, so:(1/2) * |AC| * |BD| * sin φ = 15We have |AC| = m, |BD| = n = (4/3)m, and sin φ can be expressed in terms of cos φ.Since sin^2 φ + cos^2 φ = 1, sin φ = sqrt(1 - cos^2 φ) = sqrt(1 - (36/m^2)).So, substituting into the area equation:(1/2) * m * (4/3)m * sqrt(1 - 36/m^2) = 15Simplify:(1/2) * (4/3) * m^2 * sqrt(1 - 36/m^2) = 15(2/3) * m^2 * sqrt(1 - 36/m^2) = 15Multiply both sides by 3/2:m^2 * sqrt(1 - 36/m^2) = (15 * 3)/2 = 45/2Let me denote m^2 = x for simplicity.Then, the equation becomes:sqrt(x - 36) * x = 45/2Wait, no. Wait, sqrt(1 - 36/m^2) is sqrt(1 - 36/x). So, the equation is:x * sqrt(1 - 36/x) = 45/2Let me rewrite sqrt(1 - 36/x) as sqrt((x - 36)/x) = sqrt(x - 36)/sqrt(x)So, the equation becomes:x * (sqrt(x - 36)/sqrt(x)) = 45/2Simplify:sqrt(x) * sqrt(x - 36) = 45/2Square both sides:x(x - 36) = (45/2)^2 = 2025/4Expand left side:x^2 - 36x = 2025/4Multiply both sides by 4 to eliminate the denominator:4x^2 - 144x = 2025Bring all terms to one side:4x^2 - 144x - 2025 = 0Divide the entire equation by 4 to simplify:x^2 - 36x - 506.25 = 0Wait, 2025 divided by 4 is 506.25. Hmm, dealing with decimals might complicate things. Maybe I should have kept it as fractions.Let me go back to 4x^2 - 144x - 2025 = 0Let me write 2025 as 45^2, which is 2025.So, 4x^2 - 144x - 2025 = 0Let me try to solve this quadratic equation for x.Using the quadratic formula:x = [144 ± sqrt(144^2 + 4 * 4 * 2025)] / (2 * 4)Compute discriminant:D = 144^2 + 4 * 4 * 2025144^2 = 207364 * 4 * 2025 = 16 * 2025 = 32400So, D = 20736 + 32400 = 53136Now, sqrt(53136). Let me see:230^2 = 52900, which is close. 230^2 = 52900, so 230^2 + 236 = 53136.Wait, 230^2 = 52900, so 53136 - 52900 = 236. Hmm, 236 is not a perfect square.Wait, maybe I made a mistake in calculating D.Wait, D = 144^2 + 4 * 4 * 2025144^2 = 207364 * 4 * 2025 = 16 * 2025 = 32400So, D = 20736 + 32400 = 53136Now, sqrt(53136). Let me factor 53136.Divide by 16: 53136 /16 = 33213321: sum of digits is 3+3+2+1=9, so divisible by 9.3321 /9 = 369369 /9 = 41So, 53136 = 16 * 9 * 9 * 41 = 16 * 81 * 41Thus, sqrt(53136) = 4 * 9 * sqrt(41) = 36√41So, x = [144 ± 36√41] / 8Simplify:x = (144 ± 36√41)/8 = (36(4 ± √41))/8 = (9(4 ± √41))/2Since x = m^2 must be positive, we take the positive root:x = (9(4 + √41))/2So, m^2 = (9(4 + √41))/2But m is |AC|, and n = (4/3)m, so n^2 = (16/9)m^2Thus, n^2 = (16/9) * (9(4 + √41))/2 = (16/9)*(9/2)*(4 + √41) = (16/2)*(4 + √41) = 8*(4 + √41) = 32 + 8√41Therefore, d^2 = n^2 = 32 + 8√41So, m = 32, n = 8, p = 41Thus, m + n + p = 32 + 8 + 41 = 81Wait, but the options given are 81, 89, 97, 105, 113. So, 81 is an option, which is A.But wait, let me double-check my steps because sometimes when dealing with projections, I might have mixed up something.I assumed that PQ is the projection of AC onto BD, which is |AC| cos φ = 6, and RS is the projection of BD onto AC, which is |BD| cos φ = 8. Then, using the area formula, I arrived at d^2 = 32 + 8√41, which gives m + n + p = 81.But let me think again about the projections. When you project A and C onto BD, the distance between P and Q is PQ = 6. Similarly, projecting B and D onto AC gives RS = 8.Is PQ equal to the projection of AC onto BD? Or is it something else?Wait, AC is a diagonal, so projecting A and C onto BD would give two points P and Q on BD. The distance between P and Q is PQ = 6. Similarly, projecting B and D onto AC gives R and S, with RS = 8.But in reality, the projection of AC onto BD is not necessarily equal to PQ. Instead, PQ is the distance between the projections of A and C onto BD, which might be related to the projection of vector AC onto BD.Similarly, RS is the distance between the projections of B and D onto AC, which is related to the projection of vector BD onto AC.So, actually, PQ is equal to the projection of AC onto BD, and RS is equal to the projection of BD onto AC.Wait, that makes sense because when you project two points onto a line, the distance between their projections is equal to the projection of the vector between the two points onto that line.So, PQ = |AC| cos φ = 6, and RS = |BD| cos φ = 8.Therefore, my initial assumption was correct.So, proceeding as before, |AC| cos φ = 6, |BD| cos φ = 8, and the area is (1/2)|AC||BD| sin φ = 15.So, with |AC| = m, |BD| = n, we have:m cos φ = 6n cos φ = 8Thus, n = (8/6)m = (4/3)mAnd, (1/2) m n sin φ = 15Express sin φ in terms of cos φ:sin φ = sqrt(1 - (36/m^2))Substitute into the area equation:(1/2) * m * (4/3 m) * sqrt(1 - 36/m^2) = 15Simplify:(2/3) m^2 sqrt(1 - 36/m^2) = 15Multiply both sides by 3/2:m^2 sqrt(1 - 36/m^2) = 45/2Let x = m^2:sqrt(x - 36) * sqrt(x) = 45/2Wait, no. Wait, sqrt(1 - 36/m^2) is sqrt((m^2 - 36)/m^2) = sqrt(m^2 - 36)/mSo, the equation becomes:m^2 * (sqrt(m^2 - 36)/m) = 45/2Simplify:m * sqrt(m^2 - 36) = 45/2Square both sides:m^2 (m^2 - 36) = (45/2)^2 = 2025/4So,m^4 - 36 m^2 - 2025/4 = 0Let me set y = m^2:y^2 - 36 y - 2025/4 = 0Multiply both sides by 4 to eliminate the fraction:4 y^2 - 144 y - 2025 = 0Now, solve for y:y = [144 ± sqrt(144^2 + 4 * 4 * 2025)] / (2 * 4)Compute discriminant:144^2 = 207364 * 4 * 2025 = 16 * 2025 = 32400So, discriminant D = 20736 + 32400 = 53136sqrt(53136) = 230.5... Wait, earlier I factored it as 16 * 9 * 9 * 41, so sqrt(53136) = 4 * 9 * sqrt(41) = 36√41Thus,y = [144 ± 36√41]/8 = [36(4 ± √41)]/8 = [9(4 ± √41)]/2Since y = m^2 must be positive, we take the positive root:y = [9(4 + √41)]/2Thus, m^2 = [9(4 + √41)]/2Then, n = (4/3)m, so n^2 = (16/9)m^2 = (16/9)*[9(4 + √41)]/2 = (16/9)*(9/2)*(4 + √41) = (16/2)*(4 + √41) = 8*(4 + √41) = 32 + 8√41Therefore, d^2 = n^2 = 32 + 8√41So, m = 32, n = 8, p = 41Thus, m + n + p = 32 + 8 + 41 = 81So, the answer is 81, which corresponds to option A.But wait, let me just verify once more because sometimes in these projection problems, the distance between projections can be related differently.Suppose I consider the projections of A and C onto BD. Since A and C are opposite vertices, their projections P and Q on BD should be symmetric with respect to the midpoint of BD. Therefore, the distance PQ should be twice the distance from the midpoint to P or Q.But in the problem, PQ is given as 6, which is the distance between P and Q on BD. Similarly, RS is 8.But in my earlier reasoning, I considered PQ as the projection of AC onto BD, which is |AC| cos φ = 6, and RS as |BD| cos φ = 8.Given that, and using the area formula, I arrived at d^2 = 32 + 8√41, which gives m + n + p = 81.I think this is correct. So, I'll go with 81.

answer：Okay, so I need to figure out the probability of randomly selecting a 6-digit savings card PIN where the first two digits are both 6. Each PIN has a non-zero first digit, which means the first digit can't be zero. Alright, let's break this down step by step.First, I should understand what a 6-digit PIN looks like. It's something like ABCDEF, where each letter represents a digit. The first digit, A, can't be zero, so it has to be from 1 to 9. The other digits, B, C, D, E, and F, can be any digit from 0 to 9.Now, the question is asking for the probability that both the first and second digits are 6. So, we're looking for PINs that start with 66, like 66CDEF. To find this probability, I think I need to find two things: the total number of possible 6-digit PINs with a non-zero first digit, and the number of those PINs that start with 66.Let's start with the total number of possible PINs. Since the first digit can be any from 1 to 9, there are 9 possibilities for the first digit. For each of the remaining five digits, there are 10 possibilities each (0 through 9). So, the total number of PINs should be 9 multiplied by 10 five times. That would be 9 * 10^5.Now, for the number of PINs that start with 66. If the first two digits are fixed as 6, then the first digit is 6, and the second digit is also 6. So, we don't have any choices for the first two digits; they're both determined. That leaves us with the last four digits, which can be any digit from 0 to 9. So, for each of these four positions, there are 10 possibilities. Therefore, the number of such PINs is 10^4.To find the probability, I need to divide the number of favorable outcomes (PINs starting with 66) by the total number of possible outcomes (all valid PINs). So, the probability should be 10^4 divided by 9 * 10^5.Simplifying that, 10^4 is 10,000, and 9 * 10^5 is 900,000. So, 10,000 divided by 900,000. If I divide both numerator and denominator by 10,000, that simplifies to 1/90.Wait, does that make sense? Let me double-check. The total number of PINs is 9 * 10^5, which is 900,000. The number of PINs starting with 66 is 10^4, which is 10,000. So, 10,000 divided by 900,000 is indeed 1/90. Yeah, that seems right.I think I got it. The probability is 1/90.

answer：Okay, so I'm trying to figure out this problem about the merry-go-round and the horses. Let me see if I understand it correctly. There's a horse that's 36 feet away from the center of a merry-go-round, and it makes 40 revolutions. Now, I need to find out how far another horse travels if it's only 12 feet from the center and makes 120 revolutions. Hmm, okay.First, I think I need to understand what it means for a horse to make a revolution on a merry-go-round. A revolution is like going around in a circle once, right? So, the distance a horse travels in one revolution would be the circumference of the circle it's moving along. The formula for the circumference of a circle is ( C = 2pi r ), where ( r ) is the radius, or the distance from the center.So, for the first horse that's 36 feet from the center, its circumference would be ( 2pi times 36 ). Let me calculate that: ( 2 times 3.1416 times 36 ). That would be approximately ( 226.1946 ) feet for one revolution. If it makes 40 revolutions, then the total distance it travels would be ( 40 times 226.1946 ), which is about ( 9047.784 ) feet. Okay, that makes sense.Now, the second horse is only 12 feet from the center. So, its circumference would be ( 2pi times 12 ). Let me calculate that: ( 2 times 3.1416 times 12 ) is approximately ( 75.3984 ) feet for one revolution. If this horse makes 120 revolutions, then the total distance it travels would be ( 120 times 75.3984 ), which is about ( 9047.808 ) feet. Wait a minute, that's almost the same as the first horse's distance. Is that possible?Let me double-check my calculations. For the first horse: ( 2 times 3.1416 times 36 = 226.1946 ) feet per revolution. Multiply by 40: ( 226.1946 times 40 = 9047.784 ) feet. For the second horse: ( 2 times 3.1416 times 12 = 75.3984 ) feet per revolution. Multiply by 120: ( 75.3984 times 120 = 9047.808 ) feet. Hmm, they're almost identical. That seems a bit odd, but maybe it's correct because even though the second horse is closer to the center, it's making three times as many revolutions.Let me think about it differently. Maybe there's a relationship between the radius and the number of revolutions that keeps the total distance the same. If the radius is smaller, the circumference is smaller, but if you make more revolutions, the total distance could stay the same. Let's see: the first horse has a radius of 36 feet and makes 40 revolutions. The second horse has a radius of 12 feet, which is one-third of 36 feet. If the number of revolutions increases by three times, from 40 to 120, then the total distance should remain the same because ( 40 times 36 = 120 times 12 ). Oh, that's interesting! So, ( 40 times 36 = 1440 ) and ( 120 times 12 = 1440 ) as well. So, the product of the number of revolutions and the radius is the same for both horses. That means the total distance traveled would be the same because distance is circumference times number of revolutions, and circumference is ( 2pi r ). So, ( 2pi r times text{revolutions} = 2pi times (r times text{revolutions}) ). Since ( r times text{revolutions} ) is the same for both, the total distance is the same.Wait, but in my calculations, I got slightly different numbers: 9047.784 and 9047.808. That's probably just due to rounding errors because I used an approximate value for ( pi ). If I use the exact value, they should be exactly the same. Let me try that.Using the exact value of ( pi ), the first horse's distance is ( 40 times 2pi times 36 = 2880pi ). The second horse's distance is ( 120 times 2pi times 12 = 2880pi ). Oh, so they're exactly the same! My approximate calculations had a tiny difference because I used 3.1416 for ( pi ), but symbolically, they're equal. That makes sense now.So, the distance traveled by both horses is the same, which is ( 2880pi ) feet. That's pretty cool! Even though one horse is closer to the center and makes more revolutions, the total distance ends up being the same as the horse that's farther out and makes fewer revolutions.I guess the key takeaway here is that the total distance traveled depends on both the circumference of the path and the number of revolutions. If you change one, you can adjust the other to keep the total distance the same. In this case, reducing the radius by a factor of three and increasing the number of revolutions by the same factor keeps the total distance constant.Another way to think about it is that the product of the radius and the number of revolutions is a measure of how much "work" the horse is doing in terms of moving around the merry-go-round. If you triple the number of revolutions while reducing the radius to a third, the "work" remains the same, and thus the total distance traveled remains the same.This problem also shows how proportional reasoning works in geometry. The circumference is directly proportional to the radius, and the total distance is directly proportional to both the circumference and the number of revolutions. So, by adjusting these two factors inversely, you can maintain the same total distance.I wonder if this has any real-world applications. Maybe in machinery or engineering, where different parts have to move at certain speeds or cover specific distances. Understanding how to adjust parameters like radius and revolutions could be important in designing systems that need to maintain consistent performance despite changes in individual components.Also, this problem reinforces the importance of using exact values in mathematical calculations, especially when dealing with proportions and equalities. Using approximate values can sometimes hide the true relationships between quantities, so it's better to keep things symbolic until the final step if possible.In summary, by calculating the circumference for each horse's path and then multiplying by the number of revolutions, we find that both horses travel the same total distance of ( 2880pi ) feet. This happens because the decrease in radius is exactly balanced by an increase in the number of revolutions, keeping the product ( r times text{revolutions} ) constant.