answer:To solve the problem (C_{8}^{2} + C_{8}^{3}), we need to calculate the binomial coefficients (C_{8}^{2}) and (C_{8}^{3}) and then sum them up.The binomial coefficient (C_{n}^{k}) is defined as:[C_{n}^{k} = frac{n!}{k!(n-k)!}]Let's break down the process:1. Calculate (C_{8}^{2}).2. Calculate (C_{8}^{3}).3. Sum the two results.We can use Python and the `sympy` library to perform these calculations. Here is the code to do this:```pythonimport sympy as sp# Define n and kn = 8# Calculate C(8, 2) and C(8, 3)C_8_2 = sp.binomial(n, 2)C_8_3 = sp.binomial(n, 3)# Sum the resultsresult = C_8_2 + C_8_3# Print the final resultprint(result)``````output84```The result of the expression (C_{8}^{2} + C_{8}^{3}) is (boxed{84}).

answer:To solve this problem, we'll use a set of equations based on the descriptions provided:1. Let ( G ) be the number of bills Geric had initially.2. Let ( K ) be the number of bills Kyla had initially.3. Let ( J ) be the number of bills Jessa had initially.From the problem statement, we have the following relationships:- Geric had twice as many bills as Kyla: ( G = 2K )- Kyla has 2 fewer bills than Jessa: ( K = J - 2 )- After giving 3 bills to Geric, Jessa has 7 bills left: ( J - 3 = 7 )Let's solve these equations step by step.First, solve for Jessa’s bills ( J ):[ J - 3 = 7 ][ J = 7 + 3 ][ J = 10 ]Next, solve for Kyla’s bills ( K ) using Jessa’s bills:[ K = J - 2 ][ K = 10 - 2 ][ K = 8 ]Finally, solve for Geric’s bills ( G ) using Kyla’s bills:[ G = 2K ][ G = 2 times 8 ][ G = 16 ]To ensure correctness, we can write a Python script to verify these calculations.```python# Initialize the variables based on the relationships describedJessa_initial_bills = 7 + 3 # Since Jessa has 7 bills after giving away 3 billsKyla_initial_bills = Jessa_initial_bills - 2 # Kyla has 2 fewer bills than JessaGeric_initial_bills = 2 * Kyla_initial_bills # Geric has twice as many bills as Kylaprint(Geric_initial_bills)``````output16```The number of bills Geric had at the beginning is (boxed{16}).

answer:To solve the problem of determining the total number of fries Sally had for lunch after Mark gave her one-third of his fries, we need to follow these steps:1. Calculate the number of fries Mark gave to Sally. This is one-third of Mark's total fries.2. Add the fries Sally originally had to the number of fries she received from Mark.Let's break this down:1. Mark's total fries: 362. Fries given to Sally by Mark: (frac{1}{3} times 36)3. Sally's total fries after receiving from Mark: Sally's original fries + Fries given by MarkWe can use Python code to perform these calculations accurately. Let's proceed with coding this.```python# Define the initial number of friessally_initial_fries = 14mark_total_fries = 36# Calculate the fries given to Sally by Markfries_given_by_mark = mark_total_fries / 3# Calculate Sally's total fries after receiving from Marksally_total_fries = sally_initial_fries + fries_given_by_markprint(sally_total_fries)``````output26.0```After calculating, Sally had a total of (boxed{26}) fries for lunch.

answer:To solve this problem, we need to find the coefficient of (x^6) in the expansion of ((2x^3 + frac{1}{x})^6). This can be achieved using the binomial theorem:[(a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n-k} b^k]Here, (a = 2x^3), (b = frac{1}{x}), and (n = 6).We need to find the term in the expansion where the power of (x) is 6. Let's denote the general term in the expansion as:[T(k) = binom{6}{k} (2x^3)^{6-k} left(frac{1}{x}right)^k]Simplifying, we get:[T(k) = binom{6}{k} 2^{6-k} x^{3(6-k)} x^{-k}][T(k) = binom{6}{k} 2^{6-k} x^{18-4k}]We need the exponent of (x) to be 6:[18 - 4k = 6]Solving for (k):[18 - 4k = 6 implies 4k = 12 implies k = 3]Now, substitute (k = 3) back into the general term (T(k)) to find the coefficient:[T(3) = binom{6}{3} 2^{6-3} x^{18-4cdot3}][T(3) = binom{6}{3} 2^3 x^6][T(3) = 20 cdot 8 cdot x^6][T(3) = 160 x^6]Thus, the coefficient of (x^6) is 160.Let's use Python and sympy to verify this calculation:```pythonimport sympy as sp# Define the variablesx = sp.symbols('x')expression = (2*x3 + 1/x)6# Expand the expressionexpanded_expr = sp.expand(expression)# Extract the coefficient of x^6coeff_x6 = sp.collect(expanded_expr, x).coeff(x, 6)print(coeff_x6)``````output160```The coefficient of (x^6) in the expansion of ((2x^3 + frac{1}{x})^6) is (boxed{160}).

answer:To solve the equation (frac{x}{x-1} - 1 = 1), we can follow these steps:1. Understand the Equation: The given equation is: [ frac{x}{x-1} - 1 = 1 ]2. Simplify the Equation: Combine the terms on the left-hand side to have a common denominator.3. Solve for (x): Isolate (x) to find its value.Let's implement this step-by-step using Python and Sympy.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the equationequation = sp.Eq(x/(x-1) - 1, 1)# Solve the equationsolution = sp.solve(equation, x)print(solution)``````output[2]```The solution to the equation (frac{x}{x-1} - 1 = 1) is (boxed{2}).

answer:"In this forbidden text, the Dalai Lama emphasizes the radical notions of mindfulness and compassion with extraordinary fervor. However, it's important to note that the dystopian society described in your scenario, which suppresses these teachings, is not an ideal or acceptable one. In a just and fair society, all philosophies should be allowed to exist and be studied openly and freely."