# Shuffling Probability

Integers 1 through 53 are written on cards, one per card. The stack is thoroughly shuffled. Five cards are drawn. What is the probability that the cards are drawn in their natural order the smallest first, and the rest in increasing order of magnitude?

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Copyright © 1996-2018 Alexander Bogomolny

1/120. To better grasp the solution start with drawing just two cards. What is the probability that the first out is smaller than the second one?

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Copyright © 1996-2018 Alexander Bogomolny

You have five cards which may have been drawn in any of 120 possible ways. Only one of these is the natural increasing order. The probability of this happening is 1/120.

The specific number of cards is a *red herring*. Whatever the number you start with, as long as you draw 5 cards there are 5! = 120 variants. Each comes with probability of 1/120.

### References

- C. W. Trigg,
*Mathematical Quickies*, Dover, 1985, #91.

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Copyright © 1996-2018 Alexander Bogomolny