In how many ways can the letters of the word TAMANNA be arranged ?

Options

A. a. 120
B. b. 420
C. c. 840
D. d. none of these

Answer

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

### Step 1: Count the total number of letters
The word "TAMANNA" consists of 7 letters: T, A, M, A, N, N, A.

### Step 2: Identify the frequency of each letter
- T appears 1 time
- A appears 3 times
- M appears 1 time
- N appears 2 times

### 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:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \times \ldots} \]

Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, n_3, \ldots \) are the frequencies of the repeated letters.

### Step 4: Apply the formula
In our case:
- Total letters \( n = 7 \)
- Frequencies: \( n_T = 1 \) (for T), \( n_A = 3 \) (for A), \( n_M = 1 \) (for M), \( n_N = 2 \) (for N)

Thus, the number of arrangements is:

\[ \text{Number of arrangements} = \frac{7!}{1! \times 3! \times 1! \times 2!} \]

### Step 5: Calculate the factorials
Now we calculate the factorials:
- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
- \( 1! = 1 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 2! = 2 \times 1 = 2 \)

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

\[ \text{Number of arrangements} = \frac{5040}{1 \times 6 \times 1 \times 2} = \frac{5040}{12} = 420 \]

### Step 7: Conclusion
Therefore, the total number of ways to arrange the letters of the word "TAMANNA" is **420**.

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