How many words can be formed by using 4 letters at a time out of the letters of the word MATHEMATICS?

Options

A. 2445
B. 2454
C. 1243
D. 1454

Answer

To solve the problem of how many words can be formed by using 4 letters at a time from the letters of the word "MATHEMATICS," we will follow these steps:

### Step 1: Identify the letters and their frequencies
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: Determine the unique letters
The unique letters in "MATHEMATICS" are M, A, T, H, E, I, C, S. Thus, we have a total of 8 unique letters.

### Step 3: Calculate the total arrangements for different cases
We will consider different cases based on the repetition of letters.

#### Case 1: All four letters are different
We can choose 4 different letters from the 8 unique letters. The number of ways to choose 4 letters from 8 is given by:
\[ \text{Number of ways to choose 4 letters} = \binom{8}{4} \]
After choosing 4 letters, we can arrange them in \(4!\) ways.

Calculating:
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]
\[ 4! = 24 \]
Thus, the total arrangements for this case:
\[ 70 \times 24 = 1680 \]

#### Case 2: Two letters are alike, and two letters are different
We can choose one letter that appears twice (M, A, or T), and then choose 2 different letters from the remaining 7 letters.

1. **Choose the letter that appears twice**: There are 3 options (M, A, or T).
2. **Choose 2 different letters from the remaining 7**:
\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]
3. **Arrange the letters**: The arrangement for 2 alike and 2 different letters is given by:
\[ \frac{4!}{2!} = 12 \]

Thus, the total arrangements for this case:
\[ 3 \times 21 \times 12 = 756 \]

#### Case 3: Two pairs of alike letters
We can choose 2 letters that appear twice (M, A, or T). The number of ways to choose 2 letters from the 3 pairs is:
\[ \binom{3}{2} = 3 \]
The arrangement for 2 pairs of alike letters is:
\[ \frac{4!}{2! \times 2!} = 6 \]

Thus, the total arrangements for this case:
\[ 3 \times 6 = 18 \]

### Step 4: Total arrangements
Now, we can sum up all the cases:
\[ \text{Total} = 1680 + 756 + 18 = 2454 \]

### Final Answer
The total number of words that can be formed by using 4 letters at a time from the letters of the word "MATHEMATICS" is **2454**.

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