Expected number of guesses

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What is the expected number of guesses required to obtain the word 'ANT' by randomly combining three different letters?

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I assume that there are $n = 26$ letters in your alphabet. There is $n^3 = 17576$ possibilities to draw an ordered sequence $s$ of $3$ letters. All such sequences $s$ have the same probability of being drawn, which is $p = 1/17576$. Your sequence $s = ANT$ is therefore also drawn with probability $p$. The expected number of guesses $E$ therefore is the expected number of trials until first success of a Bernoulli process with success probability $p$ and is given by $E = 1/p = 17576$.