To solve the problem of finding how many sets of two letters in the word "ARISTOCRAT" have as many letters between them as in the alphabetical order, we will follow these steps:
### Step 1: Identify the letters in "ARISTOCRAT"
The letters in the word "ARISTOCRAT" are:
- A, R, I, S, T, O, C, R, A, T
### Step 2: Determine the alphabetical order of the letters
The alphabetical order of the unique letters in "ARISTOCRAT" is:
- A, C, I, O, R, S, T
### Step 3: Identify pairs of letters
We will check pairs of letters from the word "ARISTOCRAT" and see how many letters are between them in the word compared to how many letters are between them in the alphabetical order.
### Step 4: Analyze pairs of letters
We will analyze pairs of letters systematically:
1. **A and R**:
- In "ARISTOCRAT": 0 letters between A and R.
- In Alphabetical Order: 2 letters (B, C).
- Not a valid pair.
2. **A and I**:
- In "ARISTOCRAT": 1 letter (R) between A and I.
- In Alphabetical Order: 2 letters (B, C).
- Not a valid pair.
3. **A and S**:
- In "ARISTOCRAT": 2 letters (R, I) between A and S.
- In Alphabetical Order: 3 letters (B, C, D).
- Not a valid pair.
4. **A and T**:
- In "ARISTOCRAT": 3 letters (R, I, S) between A and T.
- In Alphabetical Order: 4 letters (B, C, D, E).
- Not a valid pair.
5. **A and O**:
- In "ARISTOCRAT": 4 letters (R, I, S, T) between A and O.
- In Alphabetical Order: 3 letters (B, C, D).
- Not a valid pair.
6. **A and C**:
- In "ARISTOCRAT": 5 letters (R, I, S, T, O) between A and C.
- In Alphabetical Order: 1 letter (B).
- Not a valid pair.
7. **R and I**:
- In "ARISTOCRAT": 0 letters between R and I.
- In Alphabetical Order: 1 letter (R).
- Not a valid pair.
8. **R and S**:
- In "ARISTOCRAT": 1 letter (I) between R and S.
- In Alphabetical Order: 0 letters.
- Not a valid pair.
9. **R and T**:
- In "ARISTOCRAT": 1 letter (S) between R and T.
- In Alphabetical Order: 0 letters.
- Not a valid pair.
10. **I and S**:
- In "ARISTOCRAT": 0 letters between I and S.
- In Alphabetical Order: 1 letter (R).
- Not a valid pair.
11. **I and T**:
- In "ARISTOCRAT": 0 letters between I and T.
- In Alphabetical Order: 1 letter (R).
- Not a valid pair.
12. **S and T**:
- In "ARISTOCRAT": 0 letters between S and T.
- In Alphabetical Order: 0 letters.
- Valid pair: (S, T).
13. **C and R**:
- In "ARISTOCRAT": 1 letter (A) between C and R.
- In Alphabetical Order: 1 letter (C).
- Valid pair: (C, R).
### Step 5: Count valid pairs
From our analysis, we found the following valid pairs:
1. (S, T)
2. (C, R)
### Final Count
Thus, the total number of valid pairs is **2**.
