answer:To solve this problem, we need to express the given expression in an irreducible fraction form. Let's break it down into manageable parts and solve it step-by-step using Python and SymPy.The given expression is:[ 6 frac{16}{2015} times 9 frac{17}{2016} - 2 frac{1999}{2015} times 17 frac{1999}{2016} - 27 times frac{16}{2015} ]First, we'll convert the mixed fractions to improper fractions:- ( 6 frac{16}{2015} = frac{6 times 2015 + 16}{2015} = frac{12090 + 16}{2015} = frac{12106}{2015} )- ( 9 frac{17}{2016} = frac{9 times 2016 + 17}{2016} = frac{18144 + 17}{2016} = frac{18161}{2016} )- ( 2 frac{1999}{2015} = frac{2 times 2015 + 1999}{2015} = frac{4030 + 1999}{2015} = frac{6029}{2015} )- ( 17 frac{1999}{2016} = frac{17 times 2016 + 1999}{2016} = frac{34272 + 1999}{2016} = frac{36271}{2016} )- ( 27 times frac{16}{2015} = frac{27 times 16}{2015} = frac{432}{2015} )Next, we'll substitute these improper fractions into the original expression and perform the arithmetic operations. Let's use Python and SymPy to simplify this step-by-step.```pythonimport sympy as sp# Define the fractionsfrac1 = sp.Rational(12106, 2015)frac2 = sp.Rational(18161, 2016)frac3 = sp.Rational(6029, 2015)frac4 = sp.Rational(36271, 2016)frac5 = sp.Rational(432, 2015)# Define the expressionexpression = frac1 * frac2 - frac3 * frac4 - frac5# Simplify the expressionsimplified_expr = sp.simplify(expression)# Display the resultprint(simplified_expr)``````output17/224```The given expression simplifies to the irreducible fraction:[ boxed{frac{17}{224}} ]This is the final result.

answer:To solve this problem, we need to count the number of subsets of the set ({0, 1}). The number of subsets of a set with (n) elements is given by (2^n). In this case, the set ({0, 1}) has 2 elements. Therefore, the number of subsets is (2^2).Let's calculate it using Python and sympy:```pythonimport sympy as sp# Number of elements in the setn = 2# Calculate the number of subsetsnumber_of_subsets = 2n# Print the resultprint(number_of_subsets)``````output4```The number of subsets of the set ({0, 1}) is (boxed{4}).Here's the reasoning breakdown:- The set ({0, 1}) has 2 elements.- The number of subsets of a set with (n) elements is (2^n).- Substituting (n = 2), we get (2^2 = 4).Thus, there are 4 subsets of the set ({0, 1}).The number of subsets of the set ({0, 1}) is (boxed{4}).

answer:To solve this problem, we need to understand the properties of an arithmetic sequence and the sum of its terms.1. An arithmetic sequence can be defined by the first term (a_1) and the common difference (d). The (n)-th term of the sequence can be written as: [ a_n = a_1 + (n-1)d ]2. Given that (a_3 = 7), we can write: [ a_3 = a_1 + 2d = 7 ]3. Given that (a_1 + a_7 = 10), we can write: [ a_1 + (a_1 + 6d) = 10 implies 2a_1 + 6d = 10 ]4. The sum of the first (n) terms (S_n) of an arithmetic sequence is given by: [ S_n = frac{n}{2} (2a_1 + (n-1)d) ]5. To find the value of (n) that makes (S_n) reach its maximum value, we need to analyze (S_n) for all (n).Let's implement this step-by-step in Python to determine (a_1) and (d) first and then evaluate (S_n):```pythonimport sympy as sp# Define the symbolsa1, d = sp.symbols('a1 d')# Given equationseq1 = a1 + 2*d - 7eq2 = 2*a1 + 6*d - 10# Solve the equations to get a1 and dsolution = sp.solve((eq1, eq2), (a1, d))a1_val = solution[a1]d_val = solution[d]# Define the sum of the first n terms S_nn = sp.symbols('n')S_n = n/2 * (2*a1_val + (n-1)*d_val)# Find the value of n that maximizes S_nS_n_diff = sp.diff(S_n, n)critical_points = sp.solve(S_n_diff, n)# Filter the critical points to get the maximum valueS_n_max = max([S_n.subs(n, cp) for cp in critical_points])# Print the value of n that maximizes S_nprint("Critical Points:", critical_points)print("Maximum Sum:", S_n_max)``````outputCritical Points: [6]Maximum Sum: 36```From the output, we have determined that the critical point is (n = 6), and the maximum sum (S_n) at this point is 36.Therefore, the value of (n) that makes (S_n) reach its maximum value is (boxed{6}).The output indicates that the critical point for (n) is (6), and at this point, the sum (S_n) reaches its maximum value of (36).Therefore, the value of (n) that makes (S_n) reach its maximum value is:[boxed{6}]

answer:To determine the least number of digits in the repeating block of the fraction ( frac{8}{11} ), we can find the decimal representation of the fraction using Python and analyze it. Specifically, we are looking for the length of the repeating block.Let's go ahead and implement this.```pythonfrom decimal import Decimal, getcontext # Set a high precision to ensure we can see the repeating block getcontext().prec = 50 # Calculate 8/11 fraction = Decimal(8) / Decimal(11) # Convert the fraction to a string fraction_str = str(fraction) print(fraction_str)``````output'0.72727272727272727272727272727272727272727272727273'```The decimal representation of ( frac{8}{11} ) is ( 0.overline{72} ). This indicates that "72" is the repeating block.To ensure accuracy, let's verify this by explicitly identifying the repeating block programmatically.```python# Function to find the repeating block in the decimal expansiondef find_repeating_block(dec_str): # Remove '0.' from the start dec_str = dec_str[2:] # Loop to find the smallest repeating sequence for length in range(1, len(dec_str)): sequence = dec_str[:length] repeated = sequence * (len(dec_str) // length) if dec_str.startswith(repeated[:len(dec_str)]): return sequence return None# Verify repeating block in the fraction 8/11repeating_block = find_repeating_block(fraction_str)print((repeating_block, len(repeating_block)))``````output('7272', 4)```It seems there might be a misunderstanding in our initial approach. The detected repeating block "7272" might not be accurate since we're looking for the smallest repeating unit.Let's consider a different approach by examining the decimal representation and extracting the smallest repeating unit. Since ( frac{8}{11} = 0.overline{72} ), the repeating block should be "72".We will now focus on verifying that "72" is indeed the smallest repeating unit. Let's re-evaluate the function to ensure it correctly identifies "72" as the repeating block.```python# Check multiple lengths for repeating sequences manually # Observing the initial part of the fraction to check repeating manually beyond rounded inaccuracies test_fraction = fraction_str[2:20] # Check lengths from 1 to 10 to identify the smallest repeating sequence repeats = {} for length in range(1,11): sequence = test_fraction[:length] repeated = sequence * (len(test_fraction) // length) if test_fraction.startswith(repeated[:len(test_fraction)]): repeats[length] = sequence print(repeats)``````output{2: '72', 4: '7272', 6: '727272', 8: '72727272', 10: '7272727272'}```The manual check shows that the repeating sequence "72" occurs with different lengths, confirming that the smallest repeating unit is "72". This gives us the least number of digits in the repeating block as 2.Thus, the least number of digits in the repeating block of (frac{8}{11}) is (boxed{2}).

answer:To solve this problem, we will use the principle of inclusion-exclusion. Here's the step-by-step breakdown:1. Define the given quantities: - Total number of students, ( |U| = 62 ) - Number of students who like Chinese, ( |C| = 37 ) - Number of students who like mathematics, ( |M| = 49 ) - Number of students who like both subjects, ( |C cap M| = 30 )2. Use the principle of inclusion-exclusion: The number of students who like either Chinese or mathematics or both is given by: [ |C cup M| = |C| + |M| - |C cap M| ]3. Calculate the number of students who like neither subject: The number of students who like neither subject can be found by subtracting the number of students who like either subject or both from the total number of students: [ text{Neither} = |U| - |C cup M| ]Let's implement this in Python using sympy to ensure the result is accurate.```pythonimport sympy as sp# Define the given quantitiestotal_students = 62students_like_chinese = 37students_like_math = 49students_like_both = 30# Calculate the number of students who like either Chinese or mathematics or bothstudents_like_either_or_both = students_like_chinese + students_like_math - students_like_both# Calculate the number of students who like neither subjectstudents_like_neither = total_students - students_like_either_or_both# Print the resultprint(students_like_neither)``````output6```The number of students who like neither Chinese nor mathematics in the class is (boxed{6}).The number of students who like neither Chinese nor mathematics in the class is (boxed{6}).

answer:To solve this problem, we need to find the interior point (Z) of the triangle (ABC) that maximizes the probability (p(Z)) that the triangle formed by the lines (AA'), (BB'), and (CC') contains (Z). This requires understanding the geometric properties of the triangle and the lines formed by the segments.The point that maximizes (p(Z)) is actually well-known in geometry: it is the centroid of the triangle. The centroid is the point where the three medians of the triangle intersect, and it has several interesting properties, including the fact that it is the balance point of the triangle.Let’s find the centroid of a triangle (ABC) with vertices (A(x_1, y_1)), (B(x_2, y_2)), and (C(x_3, y_3)). The coordinates of the centroid (G) can be calculated using the formula:[G = left( frac{x_1 + x_2 + x_3}{3}, frac{y_1 + y_2 + y_3}{3} right)]Let's implement this calculation in Python using sympy.```pythonimport sympy as sp# Define symbols for vertices coordinatesx1, y1, x2, y2, x3, y3 = sp.symbols('x1 y1 x2 y2 x3 y3')# Compute the centroid of the triangleG_x = (x1 + x2 + x3) / 3G_y = (y1 + y2 + y3) / 3# Print the coordinates of the centroidcentroid = (G_x, G_y)print(centroid)``````output(x1/3 + x2/3 + x3/3, y1/3 + y2/3 + y3/3)```The output confirms that the centroid ( G ) of the triangle ( ABC ) is given by the coordinates:[G = left( frac{x_1 + x_2 + x_3}{3}, frac{y_1 + y_2 + y_3}{3} right)]This point ( G ) is the interior point of the triangle ( ABC ) that maximizes the probability ( p(Z) ) that the triangle formed by the lines ( AA' ), ( BB' ), and ( CC' ) contains ( Z ).In conclusion, the interior point ( Z ) of the triangle ( ABC ) for which ( p(Z) ) is maximized is the centroid of the triangle. The coordinates of this point are:[boxed{left( frac{x_1 + x_2 + x_3}{3}, frac{y_1 + y_2 + y_3}{3} right)}]