GATE CSE 2022 | GA Question: 5

Question

A palindrome is a word that reads the same forwards and backwards. In a game of words, a player has the following two plates painted with letters.

      

From the additional plates given in the options, which one of the combinations of additional plates would allow the player to construct a five-letter palindrome. The player should use all the five plates exactly once. The plates can be rotated in their plane.

       A. 

      B.  

      C.   

      D.  

Answer 1

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.

Comments

And the Palindrome Word is : $RADAR$

Answer 2

Answer: B

Palindrome means the string or a number that looks same from forward and backward. In a five letter word, first and fifth must be the same letters, second and fourth must have same letters. If these two are satisfied, then the middle or third letter can be any letter makes a palindrome. By observing the above given options, only option B satisfy the condition. Because the question itself does not mentioned that the order must be as given. If we can change the order according to palindrome rule. We will get option B as answer.

Answer 3

The word is RADAR