answer:We'll approach the problem step-by-step:1. List all the digits: 1, 2, and 5.2. Generate all possible three-digit permutations where all digits are different.3. Sum these numbers.Let's write the Python code to find the answer.```pythonfrom itertools import permutations# Digits to be useddigits = [1, 2, 5]# Generate all unique permutations of the three digitsperm = permutations(digits, 3)# Convert permutations to integers and calculate the sumnumbers = [100*p[0] + 10*p[1] + p[2] for p in perm]total_sum = sum(numbers)print((numbers, total_sum))``````output([125, 152, 215, 251, 512, 521], 1776)```The unique three-digit numbers formed using the digits 1, 2, and 5, where all digits are different, are: 125, 152, 215, 251, 512, and 521.The sum of all these numbers is (boxed{1776}).

answer:To solve this problem, we need to consider the conditions given and translate them into mathematical equations. Let's break down the problem:1. Let the original two-digit number be represented as (10a + b), where: - (a) is the tens digit (an integer from 1 to 9, since it's a two-digit number) - (b) is the units digit (an integer from 0 to 9)2. The number formed by swapping the tens and units digits would be (10b + a).3. According to the problem, the sum of the original number and the swapped number is 132: [ (10a + b) + (10b + a) = 132 ]Let's solve this equation step-by-step using Python and SymPy to find the values of (a) and (b) that satisfy this condition, and then count how many such pairs exist.Here's the Python code to solve the problem:```pythonimport sympy as sp# Define variablesa, b = sp.symbols('a b')# Define the equation based on the problem statementequation = (10*a + b) + (10*b + a) - 132# Solve the equation for a and bsolutions = sp.solve(equation, (a, b))# Filter out the valid solutions where a and b are integers within their respective rangesvalid_solutions = [(a_val, b_val) for a_val, b_val in solutions if 1 <= a_val <= 9 and 0 <= b_val <= 9]# Print the number of valid solutionsprint(len(valid_solutions))``````outputTraceback (most recent call last): File " valid_solutions = [(a_val, b_val) for a_val, b_val in solutions if 1 <= a_val <= 9 and 0 <= b_val <= 9] File " valid_solutions = [(a_val, b_val) f ... (output truncated)```It seems like there's an issue with how the solutions are being processed. Let's revise the approach by breaking it down more explicitly and ensuring that all steps are clear.Here’s the revised code to solve the problem:1. Define the equation based on the problem statement.2. Solve the equation.3. Iterate over the possible integer values for (a) and (b) to ensure they lie within the required ranges.4. Count the valid pairs.```pythonimport sympy as sp# Define variablesa, b = sp.symbols('a b')# Define the equation based on the problem statementequation = (10*a + b) + (10*b + a) - 132# Create a list to store valid (a, b) pairsvalid_solutions = []# Iterate through possible values for a and bfor a_val in range(1, 10): # a is the tens digit, ranges from 1 to 9 for b_val in range(0, 10): # b is the unit digit, ranges from 0 to 9 # Check if the equation holds for the given pair if equation.subs({a: a_val, b: b_val}) == 0: valid_solutions.append((a_val, b_val))# Print the number of valid solutionsprint(len(valid_solutions))``````output7```The number of valid two-digit numbers that satisfy the condition given in the problem is (boxed{7}).This means there are 7 such two-digit numbers where the sum of the number and the number formed by swapping its tens and units digits is 132.

answer:To solve the given problem (log_{2}(log_{2}16)), we need to follow these steps:1. Calculate (log_{2}16).2. Use the result from step 1 to calculate (log_{2}(text{result from step 1})).Let's break it down and implement it in Python using the `sympy` library.```pythonimport sympy as sp# Define the base and the argument for the logarithmsbase = 2# Calculate log2(16)inner_log_value = sp.log(16, base)# Calculate log2(inner_log_value)outer_log_value = sp.log(inner_log_value, base)# Display the resultprint(outer_log_value)``````output2```The output indicates that (log_{2}(log_{2}16) = 2). To verify this result:1. First, calculate (log_{2}16): [ log_{2}16 = 4 quad text{(since } 2^4 = 16text{)} ]2. Next, calculate (log_{2}4): [ log_{2}4 = 2 quad text{(since } 2^2 = 4text{)} ]Thus, (log_{2}(log_{2}16) = 2).The final answer is (boxed{2}).

answer:To solve this problem, we need to follow these steps:1. Identify the first eight prime numbers.2. Generate all unique pairs of these prime numbers.3. Calculate the sum of each pair.4. Check if the sum is a prime number.5. Count the number of pairs whose sum is prime.6. Calculate the total number of pairs.7. Determine the probability that the sum of two drawn primes is a prime number.Let's compute this step-by-step using Python:1. List the first eight prime numbers.2. Define a function to check if a number is prime.3. Iterate through all unique pairs of the prime numbers to find those whose sums are also prime.4. Calculate the final probability.Let's start by listing the first eight prime numbers and defining a function to check if a number is prime.```python# First eight prime numbersprimes = [2, 3, 5, 7, 11, 13, 17, 19]# Function to check if a number is primedef is_prime(n): if n <= 1: return False if n == 2: return True if n % 2 == 0: return False p = 3 while p * p <= n: if n % p == 0: return False p += 2 return True# Generate all unique pairs of the prime numbersfrom itertools import combinationspairs = list(combinations(primes, 2))# Check how many pairs have a sum that is primeprime_sum_pairs = [pair for pair in pairs if is_prime(sum(pair))]# Total number of pairstotal_pairs = len(pairs)# Number of pairs whose sum is primeprime_sum_pairs_count = len(prime_sum_pairs)# Probability calculationprobability = prime_sum_pairs_count / total_pairsprint((probability, prime_sum_pairs))``````output(0.14285714285714285, [(2, 3), (2, 5), (2, 11), (2, 17)])```The probability that the sum of two drawn primes is a prime number can now be determined.- There are 28 unique pairs of the first eight prime numbers.- Out of these, 4 pairs have a sum that is also a prime number: ((2, 3)), ((2, 5)), ((2, 11)), and ((2, 17)).Thus, the probability is:[frac{4}{28} = frac{1}{7}]So, the final answer is:[boxed{frac{1}{7}}]

answer:To solve the problem, we need to find ( f(f(frac{1}{27})) ) where ( f(x) ) is defined as:[ f(x) = begin{cases} log_{3}x & text{if } x > 0 left( frac{1}{2} right)^{x} & text{if } x leqslant 0 end{cases} ]Let's break down the steps to solve this:1. Evaluate ( f(frac{1}{27}) ): Since ( frac{1}{27} ) is positive, we use the first case of the function definition: [ f(frac{1}{27}) = log_{3}(frac{1}{27}) ]2. Simplify ( log_{3}(frac{1}{27}) ): Since ( frac{1}{27} = 3^{-3} ), we have: [ log_{3}(frac{1}{27}) = log_{3}(3^{-3}) = -3 ]3. Evaluate ( f(-3) ): Since (-3 leqslant 0), we use the second case of the function definition: [ f(-3) = left( frac{1}{2} right)^{-3} ]4. Simplify ( left( frac{1}{2} right)^{-3} ): [ left( frac{1}{2} right)^{-3} = 2^{3} = 8 ]Let's write the Python code using SymPy to verify these computations.```pythonimport sympy as sp# Define the function f(x)def f(x): if x > 0: return sp.log(x, 3) else: return (1/2) x# Compute f(1/27)x1 = sp.Rational(1, 27)f_x1 = f(x1)# Compute f(f(1/27))result = f(f_x1)print(result)``````output8.00000000000000```The value of ( f(f(frac{1}{27})) ) is (boxed{8}).

answer:To solve this problem, we need to set up equations based on the properties of geometric sequences.Given:- The sum of the first 5 terms of a geometric sequence is 10.- The sum of the first 10 terms is 50.We are to find the sum of the first 15 terms.1. Sum of a Geometric Sequence: The sum (S_n) of the first (n) terms of a geometric sequence with the first term (a) and common ratio (r) is given by: [ S_n = a frac{1-r^n}{1-r} quad text{if} quad r neq 1 ]2. Set Up the Equations: From the problem statement, we have two equations: [ S_5 = a frac{1-r^5}{1-r} = 10 ] [ S_{10} = a frac{1-r^{10}}{1-r} = 50 ]3. Solve for (a) and (r): We need to find (a) and (r) by solving these equations simultaneously.4. Find the Sum of the First 15 Terms: Once we have (a) and (r), we can use the formula to find (S_{15}): [ S_{15} = a frac{1-r^{15}}{1-r} ]Let's implement this in Python using sympy:```pythonimport sympy as sp# Define variablesa, r = sp.symbols('a r')# Define the equationseq1 = sp.Eq(a * (1 - r5) / (1 - r), 10)eq2 = sp.Eq(a * (1 - r10) / (1 - r), 50)# Solve the system of equations for a and rsolutions = sp.solve((eq1, eq2), (a, r))# Compute the sum of the first 15 terms using the solutions for a and ra_value = solutions[0][0]r_value = solutions[0][1]# Sum of the first 15 termsS_15 = a_value * (1 - r_value15) / (1 - r_value)print(sp.simplify(S_15))``````output210```The sum of the first 15 terms of the geometric sequence, given that the sum of the first 5 terms is 10 and the sum of the first 10 terms is 50, is (boxed{210}).