Gift Cards

A payments pipeline for an e-commerce platform must count the number of valid gift card combinations that produce an exact transaction total, supporting denomination-constrained payment flows. Determine the number of different ordered arrangements of exactly `N` gift cards (denominations: `$10`, `$30`, `$50`) to purchase an item costing `M` dollars with no remaining balance. Each card slot is distinct, so different orderings count as separate arrangements. **Input:** `N` (number of cards to use exactly) and `M` (item cost in dollars). **Output:** Integer count of valid ordered arrangements. **Key Rules:** - Must use exactly `N` cards total - Sum of all cards must equal exactly `M` - Denominations are `$10`, `$30`, `$50` (reusable) - Card slots are distinct: [$10, $30] and [$30, $10] are different arrangements - Return 0 if no valid arrangement exists