In how many different ways can the latters in the wor ''LEVEL'' be arranged?

Answer

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

### Step 1: Identify the total number of letters
The word "LEVEL" consists of 5 letters.

### Step 2: Identify the repeated letters
In the word "LEVEL":
- The letter 'L' appears 2 times.
- The letter 'E' appears 2 times.
- The letter 'V' appears 1 time.

### Step 3: Use the formula for permutations of multiset
The formula to calculate the number of distinct 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 repeated letters.

### Step 4: Apply the formula
In our case:
- \( n = 5 \) (total letters)
- The letter 'L' is repeated 2 times, so \( p_1 = 2! \)
- The letter 'E' is repeated 2 times, so \( p_2 = 2! \)
- The letter 'V' is not repeated, so we do not need to include it in the denominator.

Thus, the number of arrangements can be calculated as:

\[ \text{Number of arrangements} = \frac{5!}{2! \times 2!} \]

### Step 5: Calculate the factorials
Now we calculate the factorial values:
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Substitute the values into the formula
Now substituting the values into the formula:

\[ \text{Number of arrangements} = \frac{120}{2 \times 2} = \frac{120}{4} = 30 \]

### Conclusion
The number of different ways to arrange the letters in the word "LEVEL" is **30**.