To solve the problem of finding how many pairs of letters in the word "ADVENTURE" have the number of letters one more than the number of letters between them in the English alphabet, we can follow these steps:
### Step 1: Write down the letters of the word and their positions in the alphabet.
The letters in "ADVENTURE" are:
- A (1)
- D (4)
- V (22)
- E (5)
- N (14)
- T (20)
- U (21)
- R (18)
- E (5)
### Step 2: Identify pairs of letters.
We need to find pairs of letters (let's denote them as \(L_1\) and \(L_2\)) such that the number of letters between them in the alphabet is one less than the absolute difference of their positions.
### Step 3: Calculate the pairs.
We will check each possible pair of letters in "ADVENTURE":
1. **Pair (A, D)**:
- Positions: 1 and 4
- Letters between: 2 (B, C)
- Difference: 4 - 1 = 3 (3 is not one more than 2)
2. **Pair (A, V)**:
- Positions: 1 and 22
- Letters between: 20 (B, C, ..., U)
- Difference: 22 - 1 = 21 (21 is not one more than 20)
3. **Pair (A, E)**:
- Positions: 1 and 5
- Letters between: 3 (B, C, D)
- Difference: 5 - 1 = 4 (4 is not one more than 3)
4. **Pair (A, N)**:
- Positions: 1 and 14
- Letters between: 12 (B, C, ..., M)
- Difference: 14 - 1 = 13 (13 is not one more than 12)
5. **Pair (A, T)**:
- Positions: 1 and 20
- Letters between: 18 (B, C, ..., S)
- Difference: 20 - 1 = 19 (19 is not one more than 18)
6. **Pair (A, U)**:
- Positions: 1 and 21
- Letters between: 19 (B, C, ..., T)
- Difference: 21 - 1 = 20 (20 is not one more than 19)
7. **Pair (A, R)**:
- Positions: 1 and 18
- Letters between: 16 (B, C, ..., Q)
- Difference: 18 - 1 = 17 (17 is not one more than 16)
8. **Pair (D, V)**:
- Positions: 4 and 22
- Letters between: 17 (E, F, ..., U)
- Difference: 22 - 4 = 18 (18 is not one more than 17)
9. **Pair (D, E)**:
- Positions: 4 and 5
- Letters between: 0 (none)
- Difference: 5 - 4 = 1 (1 is one more than 0) **(Valid Pair)**
10. **Pair (D, N)**:
- Positions: 4 and 14
- Letters between: 9 (E, F, ..., I)
- Difference: 14 - 4 = 10 (10 is not one more than 9)
11. **Pair (D, T)**:
- Positions: 4 and 20
- Letters between: 15 (E, F, ..., S)
- Difference: 20 - 4 = 16 (16 is not one more than 15)
12. **Pair (D, U)**:
- Positions: 4 and 21
- Letters between: 16 (E, F, ..., T)
- Difference: 21 - 4 = 17 (17 is not one more than 16)
13. **Pair (D, R)**:
- Positions: 4 and 18
- Letters between: 13 (E, F, ..., Q)
- Difference: 18 - 4 = 14 (14 is not one more than 13)
14. **Pair (V, E)**:
- Positions: 22 and 5
- Letters between: 16 (F, G, ..., U)
- Difference: 22 - 5 = 17 (17 is not one more than 16)
15. **Pair (V, N)**:
- Positions: 22 and 14
- Letters between: 7 (W, X, Y, Z)
- Difference: 22 - 14 = 8 (8 is not one more than 7)
16. **Pair (V, T)**:
- Positions: 22 and 20
- Letters between: 1 (U)
- Difference: 22 - 20 = 2 (2 is one more than 1) **(Valid Pair)**
17. **Pair (V, U)**:
- Positions: 22 and 21
- Letters between: 0 (none)
- Difference: 22 - 21 = 1 (1 is one more than 0) **(Valid Pair)**
18. **Pair (V, R)**:
- Positions: 22 and 18
- Letters between: 3 (S, T, U)
- Difference: 22 - 18 = 4 (4 is not one more than 3)
19. **Pair (E, N)**:
- Positions: 5 and 14
- Letters between: 8 (F, G, ..., L)
- Difference: 14 - 5 = 9 (9 is not one more than 8)
20. **Pair (E, T)**:
- Positions: 5 and 20
- Letters between: 14 (F, G, ..., N)
- Difference: 20 - 5 = 15 (15 is not one more than 14)
21. **Pair (E, U)**:
- Positions: 5 and 21
- Letters between: 15 (F, G, ..., T)
- Difference: 21 - 5 = 16 (16 is not one more than 15)
22. **Pair (E, R)**:
- Positions: 5 and 18
- Letters between: 12 (F, G, ..., Q)
- Difference: 18 - 5 = 13 (13 is not one more than 12)
23. **Pair (N, T)**:
- Positions: 14 and 20
- Letters between: 5 (O, P, Q, R, S)
- Difference: 20 - 14 = 6 (6 is not one more than 5)
24. **Pair (N, U)**:
- Positions: 14 and 21
- Letters between: 6 (O, P, Q, R, S, T)
- Difference: 21 - 14 = 7 (7 is not one more than 6)
25. **Pair (N, R)**:
- Positions: 14 and 18
- Letters between: 3 (O, P, Q)
- Difference: 18 - 14 = 4 (4 is not one more than 3)
26. **Pair (T, U)**:
- Positions: 20 and 21
- Letters between: 0 (none)
- Difference: 21 - 20 = 1 (1 is one more than 0) **(Valid Pair)**
27. **Pair (T, R)**:
- Positions: 20 and 18
- Letters between: 1 (S)
- Difference: 20 - 18 = 2 (2 is one more than 1) **(Valid Pair)**
28. **Pair (U, R)**:
- Positions: 21 and 18
- Letters between: 2 (T, S)
- Difference: 21 - 18 = 3 (3 is one more than 2) **(Valid Pair)**
### Step 4: Count the valid pairs.
The valid pairs we found are:
1. (D, E)
2. (V, T)
3. (V, U)
4. (T, U)
5. (T, R)
6. (U, R)
Thus, the total number of valid pairs is **6**.
### Final Answer:
The total number of pairs of letters in the word "ADVENTURE" that satisfy the given condition is **6**.
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