Counting Rows Formed From Two Kinds of Identical Items

You have $A$ identical items of one kind and $B$ identical items of a second kind. You choose $C$ of these items and arrange them in a row. How many different rows can you form? Items of the same kind cannot be told apart, so two rows are the same exactly when they show the same kind at every position. The specific values of $A$, $B$ and $C$ are generated randomly, and the answer is an exact integer. ```hint Separate the two questions Treat which mixes of the two kinds the supplies allow separately from how many orders each allowed mix produces. ``` ### Constraints and Clarifications - $A$, $B$ and $C$ are nonnegative integers. - The order of the row matters. ### Clarifying Questions - Does order matter, or are we counting only which items are chosen? - Must every row contain at least one item of each kind? - If $C$ exceeds $A + B$, should the answer be zero? ### What a Strong Answer Covers - The feasible range for the number of first-kind items, using both supply limits - Counting the orderings for each feasible mix without double counting identical items - The simplifications when $C$ is at most both supplies and when it exceeds their total - A small hand-enumerated check ### Follow-up Questions - How does the count change with three kinds of items? - How many rows have no two adjacent items of the same kind? - If $A$, $B$ and $C$ are very large, how would you compute the answer modulo a prime efficiently?