Probability of a Norepeatword

If one must make a random Norepeatword from the 26 alphabets, what is the probability of picking a work with all 26 letters? A Norepeatword is an assemblage of any number of alphabets (1 to 26) such that no letter is repeated. Let’s do the problem step by step.

Number of 26 letter words: there are 26! words possible from 26 letters.
Number of k-letter words (k = 1 to 26): It has two parts: A) how many k-letter words are possible from 26 and B) how many rearrangements are possible for each.
A) ₂₆C_k collections can be made from 26 letters with k letters.
B) For each collection, k! rearrangements are possible.
Therefore, the number of k-letter Norepeatwords is ₂₆C_k x k!, and the total is obtained by summing from k = 1 to k = 26.

$\textrm{\# norepeatwords} = \sum\limits_{k = 1}^{26} _{26}C_k * k! = \sum\limits_{k = 1}^{26} \frac{26!}{k!(26-k)!} * k!$

The required probability is:

$\\ P = \frac{26!}{\sum\limits_{k = 1}^{26} \frac{26!}{k!(26-k)!} * k!} = \frac{26!}{\sum\limits_{k = 1}^{26} \frac{26!}{(26-k)!}} = \frac{1}{\frac{1}{25!} + \frac{1}{24!} + ... + \frac{1}{1!} + 1}$

The denominator of the equation is the famous Taylor series expansion of e^x for x = 1.
e^x = 1 + x + x²/2! + …

So, P = 1/e

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