For example, to calculate P(H_k), there are (mn)!/(mn-k)! equally likely ways to draw k balls. Of those, how many are there where the k-th ball is a horizontal win? We first select the winning k-th ball in mn ways. Then once we count the appropriate number of (unordered) selections of the previous k-1 balls using the generating function, we then order those balls in (k-1)! ways.

So we want to count unordered selections of k-1 balls, where (a) we prohibit any horizontal or vertical bingo, since we must “wait” until the k-th ball to win; and (b) we are further confined to a selections from among n-1=4 particular columns, since the initial choice of the k-th winning ball implies that the k-1 previous balls *can’t* be from that winning column; but (c) the k-1 balls must include at least a=1 from *each* of the n-1=4 other columns, since we are planning to win on the k-th draw.

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