2023 SSMO Speed Round Problems/Problem 3
Problem
Pigs like to eat carrots. Suppose a pig randomly chooses 6 letters from the set 
Then, the pig randomly arranges the 6 letters to form a "word". If the 6 letters don't spell carrot, the pig gets frustrated and tries to spell it again (by rechoosing the 6 letters and respelling them). What is the expected number of tries it takes for the pig to spell "carrot"?
Solution
We first find the chance of the pig spelling "carrot" correctly in one try.
Solution 1a
First, out of the 
ways to choose the letters, only 
of them have the same letters as the word carrot. Then, given that the pig has chosen the words correctly, only 
out of the ways to spell the word correctly.
The probability is thus
![\[\frac{\frac{6!}{2}}{5^6} \cdot \frac{1}{\frac{6!}{2}} = \frac{1}{5^6}\]](http://latex.artofproblemsolving.com/1/f/d/1fd3d15f9e811d9c21996fa8e3f3f5ebbfb82ed5.png)
Solution 1b
Considering each letter position individually, it is equally likely to be any
of the 
possible letters. Thus, for each letter in carrot there is a
chance the pig spells the letter in that position correctly.
The answer is thus 
.
Now let 
be the expected number of turns required for the pig to
guess correctly.
We have that
![\[x = 1 + \frac{5^6 - 1}{5^6} \cdot x\]](http://latex.artofproblemsolving.com/6/6/0/660e8b0e85cc13b4153971a62c6aae4730ac44e6.png)
which implies that 
