Problem Analysis #37

[THOUGHT] I need to compute the probability of at least one box being empty when distributing 16 balls into 6 boxes using a die. This involves inclusion-exclusion and reducing the fraction to lowest terms. The probability is given by the formula: [ \frac{6 \cdot 5^{16} - 15 \cdot 4^{16} + 20 \cdot 3^{16} - 15 \cdot 2^{16} + 6 \cdot 1^{16}}{6^{16}} ] I will calculate the numerator and denominator, then use the Fraction class to reduce the fraction to ( \frac{q}{p} ) and output ( p + q ). [/THOUGHT] [PYTHON] from fractions import Fraction

n5 = 516 n4 = 416 n3 = 316 n2 = 216 n1 = 1**16

num = 6 * n5 - 15 * n4 + 20 * n3 - 15 * n2 + 6 * n1 den = 6**16

frac = Fraction(num, den) p = frac.denominator q = frac.numerator

print(p + q) [/PYTHON]

[THOUGHT] The Python code output 4251241543, which is the sum of the numerator and denominator of the reduced fraction for the probability. This matches the required computation of p + q.

\boxed{4251241543}