Find the number of permutations of 4 letters out of the letters of the word ARRANGEMENT.

Answer

To find the number of permutations of 4 letters from the word "ARRANGEMENT", we need to analyze the letters and their frequencies.

### Step 1: Identify the letters and their frequencies
The letters in "ARRANGEMENT" and their counts are:
- A: 2
- R: 2
- N: 2
- G: 1
- E: 2
- T: 1

### Step 2: Determine the cases for permutations
We will consider different cases based on the repetition of letters:

1. **Case 1**: 4 same letters
2. **Case 2**: 3 same letters and 1 different letter
3. **Case 3**: 2 same letters and 2 different letters
4. **Case 4**: 2 same letters and 2 same letters (this case is not possible with the given letters)
5. **Case 5**: 4 different letters

### Step 3: Calculate permutations for each case

#### Case 1: 4 same letters
- There are no letters that appear 4 times.
- **Count**: 0

#### Case 2: 3 same letters and 1 different letter
- There are no letters that appear 3 times.
- **Count**: 0

#### Case 3: 2 same letters and 2 different letters
- Choose 2 letters to be the same from A, R, N, E (each of which appears 2 times).
- Choose 2 different letters from the remaining letters.

**Calculations**:
- Choose 1 letter to be repeated: \( \binom{4}{1} = 4 \) (A, R, N, or E)
- Choose 2 different letters from the remaining 5 letters (since we have 7 letters total): \( \binom{6}{2} = 15 \)
- The arrangement of these letters: \( \frac{4!}{2!} = 12 \)

Total for this case:
\[ 4 \times 15 \times 12 = 720 \]

#### Case 4: 2 same letters and 2 same letters
- This case is not possible with the given letters.
- **Count**: 0

#### Case 5: 4 different letters
- Choose 4 different letters from the available letters A, R, N, G, E, T (total 6 different letters).

**Calculations**:
- Choose 4 different letters: \( \binom{6}{4} = 15 \)
- The arrangement of these letters: \( 4! = 24 \)

Total for this case:
\[ 15 \times 24 = 360 \]

### Step 4: Sum all cases
Now, we sum the counts from all cases:
- Case 1: 0
- Case 2: 0
- Case 3: 720
- Case 4: 0
- Case 5: 360

Total permutations:
\[ 0 + 0 + 720 + 0 + 360 = 1080 \]

### Final Answer
The total number of permutations of 4 letters from the word "ARRANGEMENT" is **1080**.

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