A standard 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 cards. If we shuffle such a deck and draw cards without replacement, then we would need, on average, approximately cards so that we have at least one card for each rank.
Now, assume you have such decks, each with a different back design. We shuffle all 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.
一副标准的 张的扑克牌有 种花色，每种花色中有 张不同点数的牌。但是当今的 张的扑克牌里还会有大王、小王两张牌，这两张牌没有花色与点数。如果我们将这副 张牌打乱、洗净，然后一张一张摸牌，那我们期望要抽约 才能集齐所有点数。
现在，假设你有 副这样的牌，每一副牌的背面设计都不同。我们将所有 张牌打乱、洗净后一张一张摸牌，那么，如果想要集齐每个花色、每个点数、每个背面设计，期望要抽多少张牌？将你的答案四舍五入至小数点后 位。
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