In how many different ways can the letters of the word 'ABILITY' be arranged?

Options

A. 5040
B. 720
C. 1260
D. None of these

Answer

To find the number of different ways to arrange the letters of the word "ABILITY," we can follow these steps:

### Step 1: Identify the total number of letters
The word "ABILITY" consists of 7 letters.

### Step 2: Identify any repeating letters
In the word "ABILITY," the letter 'I' appears twice, while all other letters (A, B, L, T, Y) appear once.

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of letters in a word where some letters are repeated is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

Where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeating letters.

### Step 4: Apply the formula
Here, we have:
- \( n = 7 \) (total letters),
- The letter 'I' repeats 2 times, so \( p_1 = 2! \).

Thus, the number of arrangements is:

\[ \text{Number of arrangements} = \frac{7!}{2!} \]

### Step 5: Calculate the factorials
Now, we calculate \( 7! \) and \( 2! \):

- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Substitute the values into the formula
Now substitute these values into the formula:

\[ \text{Number of arrangements} = \frac{5040}{2} = 2520 \]

### Conclusion
Therefore, the total number of different ways to arrange the letters of the word "ABILITY" is **2520**.

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