Hi so i'm working out a problem from combinatorics where it says to solve for probability of a palindrom using n=7 letter word.
now the solution says to pick 4 letters and arrange them as (1,2,3,4,3,2,1). however I can form a palindrome using 2 letters correct? (1,1,1,2,1,1,1) ? If this is the okay, with repeat letters, then the problem is more complicated?
is this the right definition of a palindrom? can't i have more than 1 letter drawn?
$\endgroup$ 21 Answer
$\begingroup$There are $26^7$ possible $7$ letter words that can be formed using the alphabet [a-z].
A palindrome of length $7$ has the form: $abcdcba$, where $a,b,c,d$ do not need to be distinct, so you can choose a letter $\in$ [a-z] for each one. Thus, there are $26^4$ total palindromes of length $7$.
The probability of choosing one is:$$ \frac{26^4}{26^7} = \frac{1}{26^3} \approx 0.0000569 $$
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