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If we choose 5 cards from deck of 52, and card numbers go from 1 to 13... What is the probability that each card will have different number?

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Well, you know the sample space is 52 choose 5. Count the number of $5$ card hands that all have different numbers. You have a set of $13$ numbers, and you must choose $5$. Then, each card may take on one of four suits, so multiply $13$ choose $5$ by $4^5$, to obtain the numerator.

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$$\frac{52*48*44*40*36}{52*51*50*49*48}$$ The first draw is valid with probability $\frac{52}{52}$, afterwards there are $52-4$ valid cards of $51$ total remaining etc.

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Another approach to this can be- You have cards numbered from 1 to 13. You choose 5 numbers out of 13, i.e. (13*C*5). Now each of the chosen number has 4 types. So pick 1 from each i.e. (4*C*1) Total 5 cards drawn so (4*C*1)*5.

So final answer = (13C5)*[(4C1)*5]

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