Problem
COM-B1-M01-P021 Six-Digit Numbers with Even Digit Sum
#21
★★★★☆ Level 4 of 5
How many six-digit numbers with distinct digits have even digit sum?
Choose a digit set with an even number of odd digits, then handle the forbidden leading zero.
There are \(5\) even and \(5\) odd decimal digits. For six digits, the digit sum is even if the number of odd digits is \(2\) or \(4\). With \(2\) odd digits, there are \(10\) sets without zero and \(40\) with zero; contribution \(10\cdot720+40\cdot600=31200\). With \(4\) odd digits, there are \(30\) sets without zero and \(20\) with zero; contribution \(30\cdot720+20\cdot600=33600\). Total: \(64800\).
Regional-style for the first module: several layers of casework.