Which of the following words can be written in 120 different ways ?

Options

A. STABLE
B. STILL
C. WATER
D. NOD

Explanation

To determine which of the given words can be arranged in 120 different ways, we will calculate the number of arrangements (permutations) for each word. The formula for permutations of a word with distinct letters is given by \( n! \) (n factorial), where \( n \) is the number of letters in the word. If there are repeated letters, the formula is modified to \( \frac{n!}{p_1! \times p_2! \times \ldots} \), where \( p_1, p_2, \ldots \) are the frequencies of the repeated letters.

Let's analyze each option step by step:

### Option 1: Stable
1. Count the letters: "Stable" has 6 letters.
2. Calculate permutations: \( 6! = 720 \).
3. Conclusion: 720 is not equal to 120, so this option is incorrect.

### Option 2: Still
1. Count the letters: "Still" has 5 letters.
2. Identify repeated letters: The letter 'L' appears twice.
3. Calculate permutations:
\[ \text{Permutations} = \frac{5!}{2!} = \frac{120}{2} = 60. \]
4. Conclusion: 60 is not equal to 120, so this option is incorrect.

### Option 3: Water
1. Count the letters: "Water" has 5 letters.
2. Identify repeated letters: There are no repeated letters.
3. Calculate permutations: \( 5! = 120 \).
4. Conclusion: 120 is equal to 120, so this option is correct.

### Option 4: None
1. This option is simply stating that none of the previous options can be arranged in 120 different ways.
2. Since we found that "Water" can be arranged in 120 ways, this option is incorrect.

### Final Conclusion:
The word that can be written in 120 different ways is **Water** (Option 3).

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