Logic :
Player needs to make a 5-Letter Palindrome word.
In any odd-letters palindrome word, the middle letter must be present odd number of times. Every other letter(which is Not the middle letter) must be present even number of times.
Using this logic, we can see that in a 5-letter word, Exactly one letter must appear odd number of times(this letter will become the middle letter), and every other letter must appear even number of times.
Also, if we have the following collection of letters then we can definitely make some palindrome from them :
“Exactly one letter appears odd number of times(this letter will become the middle letter), and every other letter appears even number of times”.
So, we have a “if and only if theorem” from these observations :
$\color{blue}{\text{“Let S be a collection of letters. There exists a palindrome P using all the letters of S }}$
$\color{blue}{\text{if and only if Exactly one letter in S appears odd number of times,}}$
$\color{blue}{\text{and every other letter appears even number of times” }}$
We get answer as Option B.
Option A :
In $\{ A,D,D,D,J \}$, three letters are appearing odd number of times, So, Option A is wrong.
Option C :
In $\{ A,D,Z,D,E \}$, three letters are appearing odd number of times, So, Option C is wrong.
Option D :
In $\{ A,D,I,L,Y \}$, five letters are appearing odd number of times, So, Option D is wrong.