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Suppose we have a collection of $5,000$ standard playing cards (assuming each card is equally likely to have any suit or rank), and suppose that we randomly sort these cards into $1,000$ five-card hands. What is the probability that there is no hand in which all cards are of the same rank?

My attempt: I have tried to find the probability that in any given hand, the probability that the cards are not all of the same rank is given by $1-P(\text{all cards have same rank}) = 1-\frac{13}{13^5}$, but I'm not sure how to deal with issues relating to independence among the different hands etc.

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Because the deck is a random collection of cards, you are essentially drawing from an infinite deck or drawing from one suit with replacement. As such, all the hands are independent, so you can just raise that to the $1000$ power.

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