2024 Mock AMC 12 - Problem 15

Define the number complement of a positive integer n in base b as a new number (n')_b, where the sum of the corresponding digits of n and n' is equal to b-1. For example, the number complement of (126)_9 is (762)_9 because 1 + 7 = 2 + 6 = 6 + 2 = 8. Find the number of base 8 positive integers k with a digit sum of 32 and each digit d \in \{1,2,3,4,5,6\}, such the digit sum of k's number complement is equal to 10.

\textbf{(A) }126\qquad\textbf{(B) }210\qquad\textbf{(C) }84\qquad\textbf{(D) }252\qquad\textbf{(E) }462