Problem 796

A Grand Shuffle

A standard $52$ card deck comprises thirteen ranks in four suits. However, modern decks have two additional Jokers, which neither have a suit nor a rank, for a total of $54$ cards. If we shuffle such a deck and draw cards without replacement, then we would need, on average, approximately $29.05361725$ cards so that we have at least one card for each rank.

Now, assume you have $10$ such decks, each with a different back design. We shuffle all $10 \times 54$ cards together and draw cards without replacement. What is the expected number of cards needed so every suit, rank and deck design have at least one card?

Give your answer rounded to eight places after the decimal point.

大洗牌

一副标准扑克牌包含 $52$ 张点数牌，分为四个花色、十三种点数，另外还会有两张鬼牌（大小王），鬼牌不计花色和点数，共计 $54$ 张牌。如果我们随机洗牌并不放回地抽牌，我们需要抽平均约 $29.05361725$ 张牌才能抽齐每一种点数。

现在假设你有 $10$ 副牌，每一副牌的背面花纹都不同。我们将全部共 $10 \times 54$ 张扑克牌随机洗开，然后不放回地抽牌。为了抽齐每一种花色、点数和背面花纹，平均需要抽多少张牌？

将你的答案保留小数点后八位。