How many different 5-letter words can be formed from the word ORANGE using each letter only once?

Options

A. `._(6)P_(6)`
B. `36`
C. `._(6)C_(6)`
D. `._(6)P_(5)`

Answer

To solve the problem of how many different 5-letter words can be formed from the word "ORANGE" using each letter only once, we can follow these steps:

### Step 1: Identify the total number of letters
The word "ORANGE" consists of 6 different letters: O, R, A, N, G, E.

### Step 2: Determine the number of letters to choose
We need to form 5-letter words, so we will be selecting 5 letters from the 6 available letters.

### Step 3: Use the permutation formula
The number of arrangements (or permutations) of n items taken r at a time is given by the formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
Where:
- \( n \) is the total number of items (in this case, 6 letters),
- \( r \) is the number of items to arrange (in this case, 5 letters).

### Step 4: Substitute the values into the formula
Here, \( n = 6 \) and \( r = 5 \). Therefore, we need to calculate:
\[ P(6, 5) = \frac{6!}{(6 - 5)!} = \frac{6!}{1!} \]

### Step 5: Calculate the factorials
Now, we calculate \( 6! \) and \( 1! \):
- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)
- \( 1! = 1 \)

### Step 6: Complete the calculation
Now substitute back into the permutation formula:
\[ P(6, 5) = \frac{720}{1} = 720 \]

### Conclusion
Thus, the total number of different 5-letter words that can be formed from the letters of the word "ORANGE" is **720**.

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