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8. show that the probability that all permutations of the sequence 1,2,…,n have no number being...

8. show that the probability that all permutations of the sequence 1,2,…,n have no number being still in the ith position is less than 0, 37 if n is large enough.

Math   Statistics-And-Probability   
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Answer #1

This is question on dearrangement

In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no fixed points.

the number of permutation of n distinct object is n!

hence required probability is Dn/n!

Now Dn = n!/e as n tend to infinity

hence

required probability = 1/e = 0.3678794411 < 0.37

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