The number of permutations that can be formed from all the letters of the word 'BASEBALL' is `:`

Options

A. 540
B. 1260
C. 3780
D. 5040

Answer

To find the number of permutations that can be formed from all the letters of the word "BASEBALL", we need to follow these steps:

### Step 1: Identify the letters and their frequencies
The word "BASEBALL" consists of the following letters:
- B: 2
- A: 2
- S: 1
- E: 1
- L: 2

### Step 2: Calculate the total number of letters
Count the total number of letters in the word "BASEBALL":
- Total letters = 2 (B) + 2 (A) + 1 (S) + 1 (E) + 2 (L) = 8 letters

### Step 3: Use the formula for permutations of multiset
The formula for the number of permutations of a multiset is given by:
\[ \text{Number of permutations} = \frac{n!}{n_1! \times n_2! \times n_3! \times \ldots} \]
where \( n \) is the total number of items, and \( n_1, n_2, n_3, \ldots \) are the frequencies of the repeated items.

### Step 4: Apply the formula
In our case:
- \( n = 8 \) (total letters)
- The frequencies are:
- B: 2
- A: 2
- S: 1
- E: 1
- L: 2

Thus, we can write:
\[ \text{Number of permutations} = \frac{8!}{2! \times 2! \times 1! \times 1! \times 2!} \]

### Step 5: Calculate the factorials
Now we calculate the factorials:
- \( 8! = 40320 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

Substituting these values into the formula:
\[ \text{Number of permutations} = \frac{40320}{2 \times 2 \times 1 \times 1 \times 2} = \frac{40320}{8} \]

### Step 6: Perform the division
Now, we perform the division:
\[ \frac{40320}{8} = 5040 \]

### Final Answer
The number of permutations that can be formed from all the letters of the word "BASEBALL" is **5040**.

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