Permutations & Combinations Question - None

Question

Let $a, b, c, d$ be non-zero distinct digits. The number of 4-digit numbers $abcd$ such that $ab + cd$ is even is divisible by:

Accepted Answer

Here $ab$ denotes the two-digit number $10a + b$ and $cd$ denotes $10c + d$.

For $ab + cd$ to be even, both $ab$ and $cd$ must have the same parity (both even or both odd).

A two-digit number is even if its units digit is even, and odd if its units digit is odd.

Count the valid selections of distinct non-zero digits $a, b, c, d$ satisfying the parity condition. The total count is divisible by 3, 4, and 11.

Question: