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Find the number of palindrome words formed by taking $2$ or more letters from the word ‘RESONANCE’ (palindrome words are those which read the same forward and backward).
A) $87$
B) $57$
C) $84$
D) $26$

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Last updated date: 16th Jun 2024
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Answer
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Hint:
For palindromes, words must be the same when read forward and backward. So first we have to check for the repeating letters. Here there are two repeating letters. So we can find combinations of two, three, four and five.

Complete step by step solution:
Number of letters in the given word ‘RESONANCE’ is $9$.
We can see that two letters N and E are occurring two times and the rest of the five letters R,S,O,A and C are occurring one time.
We are asked to find the number of palindrome words formed by taking $2$ or more letters at a time.
We can consider two letter words first.
There are only two possibilities: NN and EE.
Now consider three letter words.
Possibilities are NRN, NEN, NSN, NON, NAN, NCN, ERE, ESE, EOE, ENE, EAE, ECE.
There is $12$ in total.
Now consider four letter words.
There are only two cases: NEEN and ENNE
Now consider five letter words.
Possibilities are NEREN, NESEN, NEOEN, NEAEN, NECEN, ENRNE, ENSNE, ENONE, ENANE, ENCNE.
There is $10$ in total.
In total there are $2 + 12 + 2 + 10 = 26$ possibilities.

$\therefore $ The answer is option D.

Note:
To find the words choose the repeating letters and fix them in the beginning and end tail. Then we can make different choices for the middle letter. Considering all possibilities we get the answer.