The number of words that can be formed by using the letters of the word 'MATHEMATICS' that start as well as end with T, is

Options

A. 80720
B. 90720
C. 20860
D. 37528

Answer

To solve the problem of finding the number of words that can be formed using the letters of the word "MATHEMATICS" that start and end with the letter 'T', we can follow these steps:

### Step 1: Identify the letters in "MATHEMATICS"
The word "MATHEMATICS" consists of the following letters:
- M: 2
- A: 2
- T: 2
- H: 1
- E: 1
- I: 1
- C: 1
- S: 1

### Step 2: Fix the positions of 'T'
Since the words must start and end with 'T', we can fix 'T' in the first and last positions. This means we have the following arrangement:

**T _ _ _ _ _ _ _ _ T**

We now have 9 positions left to fill with the remaining letters.

### Step 3: Count the remaining letters
After placing the two 'T's, the remaining letters are:
- M: 2
- A: 2
- H: 1
- E: 1
- I: 1
- C: 1
- S: 1

This gives us a total of 9 letters to arrange.

### Step 4: Calculate the number of arrangements
The number of ways to arrange these 9 letters, taking into account the repetitions of 'M' and 'A', can be calculated using the formula for permutations of multiset:

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

Where:
- \( n \) is the total number of letters to arrange (which is 9),
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

In our case:
- Total letters (n) = 9
- Repeated letters: M (2), A (2)

Thus, the formula becomes:

\[ \text{Number of arrangements} = \frac{9!}{2! \cdot 2!} \]

### Step 5: Calculate \( 9! \) and \( 2! \)
Calculating \( 9! \):

\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \]

Calculating \( 2! \):

\[ 2! = 2 \times 1 = 2 \]

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

\[ \text{Number of arrangements} = \frac{362880}{2 \times 2} = \frac{362880}{4} = 90720 \]

### Final Answer
The total number of words that can be formed by using the letters of the word "MATHEMATICS" that start and end with 'T' is **90720**.

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