How many pairs of letters are there in the word 'DESIGNATION' which have as many letters between them as in the English alphabetical series (both in forward and backward direction)... How many pairs of letters are there in the word 'DESIGNATION' which have as many letters between them as in the English alphabetical series (both in forward and backward direction)? Read more

Steps to Solve

1. Identify the Positions of Letters in the Alphabet

   The first step is to convert the letters in the word 'DESIGNATION' to their corresponding positions in the English alphabet:

   • D = 4
   • E = 5
   • S = 19
   • I = 9
   • G = 7
   • N = 14
   • A = 1
   • T = 20
   • I = 9
   • O = 15
   • N = 14

2. Calculate the Index of Each Pair

   For each pair of letters in 'DESIGNATION', calculate the number of letters between them. This can be calculated by taking the absolute difference of their indices:

   $$ |i - j| - 1 $$

   where $i$ and $j$ are the indices of the letters in the word.

3. Determine the Required Number of Letters Between the Pairs

   For each pair, determine if the absolute difference between their positions in the alphabet matches the number of letters between them. The condition can be represented as:

   $$ | \text{position of letter1} - \text{position of letter2} | = |i - j| - 1 $$

4. Count Valid Pairs

   Loop through all unique pairs of letters in 'DESIGNATION' and apply the above condition. Count how many pairs satisfy this condition in both forward and backward directions.