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 Letters and Their Positions

   • In the word "DESIGNATION", we have the following letters with their respective 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. Check Pairs for Forward and Backward Conditions

   • For each pair $(L_1, L_2)$ of letters at positions $(i, j)$, calculate the number of letters between them: $|j - i| - 1$.
   • Determine the positions in the alphabet: $P_1$ for $L_1$ and $P_2$ for $L_2$.

3. Forward Check

   • Check if $|j - i| - 1 = |P_2 - P_1|$ for forwards.
   • Iterate through all pairs of letters in "DESIGNATION".

4. Backward Check

   • Similarly, check if $|j - i| - 1 = |P_1 - P_2|$ for backwards.
   • This will be the same checks as the forwards since absolute differences are symmetric.

5. Count Valid Pairs

   • Maintain a counter for valid pairs that meet either the forward or backward condition.