from math import comb

def binomial_coefficient(n, k):
  return comb(n, k)

binomial_coefficient(8, 2) # 28

def hamming_distance(a, b):
  return bin(a ^ b).count('1')

hamming_distance(2, 3) # 1

def prime_factors(num):
  factors = []
  factor = 2

  while (num >= 2):
    if (num % factor == 0):
      factors.append(factor)
      num = num / factor
    else:
      factor += 1
  return factors

prime_factors(12) # [2, 2, 3]
prime_factors(42) # [2, 3, 7]

# Calculate probability of getting exactly 2 heads in 5 coin flips
total_outcomes = 2 ** 5  # 32
favorable_outcomes = binomial_coefficient(5, 2)  # 10
probability = favorable_outcomes / total_outcomes  # 0.3125 or 31.25%

original = 0b1010  # Binary: 1010
received = 0b1011  # Binary: 1011
errors = hamming_distance(original, received)  # 1

if errors > 0:
    print(f"Detected {errors} bit error(s)")

number = 15
factors = prime_factors(number)  # [3, 5]
print(f"{number} = {' × '.join(map(str, factors))}")
# Output: 15 = 3 × 5

# How many ways to choose 3 team members from 10 people?
ways = binomial_coefficient(10, 3)  # 120

# How many poker hands (5 cards from 52)?
hands = binomial_coefficient(52, 5)  # 2,598,960

pattern1 = 0b11110000
pattern2 = 0b11111111
differences = hamming_distance(pattern1, pattern2)  # 4

# Determine if patterns are similar enough
threshold = 2
if differences <= threshold:
    print("Patterns are similar")
else:
    print("Patterns are significantly different")

def is_prime(n):
    factors = prime_factors(n)
    return len(factors) == 1 and factors[0] == n

is_prime(17)  # True
is_prime(18)  # False

# Find the greatest common divisor using prime factors
def common_factors(a, b):
    factors_a = prime_factors(a)
    factors_b = prime_factors(b)
    return set(factors_a) & set(factors_b)

common_factors(12, 18)  # {2, 3}