How many independent words can be formed without changing the order of the letters and using each letter only once? CAPABILITY

Options

A. One
B. Two
C. Three
D. four

Explanation

To solve the problem of how many independent words can be formed from the letters of the word "CAPABILITY" without changing the order of the letters and using each letter only once, we can follow these steps:

### Step 1: Identify the letters in the word "CAPABILITY"
The word "CAPABILITY" consists of the following letters: C, A, P, A, B, I, L, I, T, Y.

### Step 2: Count the total letters and identify repetitions
In "CAPABILITY", we have:
- Total letters = 10
- The letter 'A' appears 2 times.
- The letter 'I' appears 2 times.

### Step 3: Determine the formula for counting arrangements
Since we need to find the number of independent arrangements (words) that can be formed while maintaining the order of the letters, we can use the concept of combinations. However, since we are not allowed to change the order of the letters, we can only consider the distinct segments that can be formed from the letters.

### Step 4: Identify the independent segments
The independent segments that can be formed from "CAPABILITY" while maintaining the order are:
1. "C"
2. "CA"
3. "CAP"
4. "CAPA"
5. "CAPAB"
6. "CAPABI"
7. "CAPABIL"
8. "CAPABILIT"
9. "CAPABILITY"

### Step 5: Count the independent segments
From the above segments, we can see that:
- The segments can be counted as 1 for each letter added to the previous segment, leading to a total of 10 segments.

### Conclusion
Thus, the total number of independent words that can be formed from the word "CAPABILITY" without changing the order of the letters is **10**.

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