Number of ways in which the letters of the word TAMANNA be arranged is

Options

A. 120
B. 420
C. 840
D. 468

Answer

To find the number of ways in which the letters of the word "TAMANNA" can be arranged, we can follow these steps:

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

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

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

In our case:
- \( n = 7 \) (total letters)
- \( p_1 = 3! \) (for A)
- \( p_2 = 2! \) (for N)

### Step 4: Plug in the values into the formula
\[ \text{Number of arrangements} = \frac{7!}{3! \times 2!} \]

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

### Step 6: Substitute the factorial values into the formula
\[ \text{Number of arrangements} = \frac{5040}{6 \times 2} = \frac{5040}{12} \]

### Step 7: Perform the division
\[ \text{Number of arrangements} = 420 \]

### Conclusion
The number of ways in which the letters of the word "TAMANNA" can be arranged is **420**.

---