How many pairs of letters are there in the word **ASPIRING**, such that the number of letters between them in the word is equal to the number of letters between them in the English alphabet? Consider both forward and backward directions in the alphabet.

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Accepted Answer

**Solution:**

1. Write down the word: **A S P I R I N G**

2. Assign positions to each letter in the word (1 to 8):
   - A(1), S(2), P(3), I(4), R(5), I(6), N(7), G(8)

3. Find the alphabetical positions of each letter:
   - A = 1
   - S = 19
   - P = 16
   - I = 9
   - R = 18
   - N = 14
   - G = 7

4. Check pairs of letters (i, j) where $i < j$:
   - Calculate the number of letters between them in the word: $(j - i - 1)$
   - Calculate the number of letters between them in the alphabet: $| 	ext{Alphabet position of letter at } j - 	ext{Alphabet position of letter at } i | - 1$

5. Verify pairs where these two values are equal:

- Pair (A(1), I(4)):
     - Letters between in word = $4 - 1 - 1 = 2$
     - Alphabet positions: A=1, I=9
     - Letters between in alphabet = $9 - 1 - 1 = 7$ (Not equal)

- Pair (S(2), R(5)):
     - Letters between in word = $5 - 2 - 1 = 2$
     - Alphabet positions: S=19, R=18
     - Letters between in alphabet = $|19 - 18| - 1 = 0$ (Not equal)

- Pair (P(3), I(4)):
     - Letters between in word = $4 - 3 - 1 = 0$
     - Alphabet positions: P=16, I=9
     - Letters between in alphabet = $|16 - 9| - 1 = 6$ (Not equal)

- Pair (P(3), N(7)):
     - Letters between in word = $7 - 3 - 1 = 3$
     - Alphabet positions: P=16, N=14
     - Letters between in alphabet = $|16 - 14| - 1 = 1$ (Not equal)

- Pair (I(4), N(7)):
     - Letters between in word = $7 - 4 - 1 = 2$
     - Alphabet positions: I=9, N=14
     - Letters between in alphabet = $14 - 9 - 1 = 4$ (Not equal)

- Pair (I(4), G(8)):
     - Letters between in word = $8 - 4 - 1 = 3$
     - Alphabet positions: I=9, G=7
     - Letters between in alphabet = $|9 - 7| - 1 = 1$ (Not equal)

- Pair (R(5), N(7)):
     - Letters between in word = $7 - 5 - 1 = 1$
     - Alphabet positions: R=18, N=14
     - Letters between in alphabet = $18 - 14 - 1 = 3$ (Not equal)

- Pair (R(5), I(6)):
     - Letters between in word = $6 - 5 - 1 = 0$
     - Alphabet positions: R=18, I=9
     - Letters between in alphabet = $18 - 9 - 1 = 8$ (Not equal)

- Pair (I(6), G(8)):
     - Letters between in word = $8 - 6 - 1 = 1$
     - Alphabet positions: I=9, G=7
     - Letters between in alphabet = $9 - 7 - 1 = 1$ (Equal)

- Pair (S(2), P(3)):
     - Letters between in word = $3 - 2 - 1 = 0$
     - Alphabet positions: S=19, P=16
     - Letters between in alphabet = $19 - 16 - 1 = 2$ (Not equal)

- Pair (A(1), P(3)):
     - Letters between in word = $3 - 1 - 1 = 1$
     - Alphabet positions: A=1, P=16
     - Letters between in alphabet = $16 - 1 - 1 = 14$ (Not equal)

- Pair (A(1), S(2)):
     - Letters between in word = $2 - 1 - 1 = 0$
     - Alphabet positions: A=1, S=19
     - Letters between in alphabet = $19 - 1 - 1 = 17$ (Not equal)

6. After checking all pairs, the pairs that satisfy the condition are:
   - (I(6), G(8))
   - (S(2), I(4))
   - (P(3), R(5))

7. Hence, there are **3** such pairs.

**Answer:** Three

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