Number of all permutations of the set of four letters E , O , S , P taken two at a time is

Options

A. `""^(4)C_(2)xx 2!`
B. `""^(4)C_(2)`
C. `2!`
D. `4!`

Answer

To find the number of all permutations of the set of four letters E, O, S, P taken two at a time, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Letters**:
We have the letters E, O, S, and P. This gives us a total of 4 letters.

2. **Select 2 Letters from the 4 Letters**:
We need to select 2 letters from the 4 available letters. The number of ways to choose 2 letters from 4 can be calculated using combinations:
\[ \text{Number of ways to choose 2 letters} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

3. **Arrange the Selected Letters**:
After selecting 2 letters, we need to arrange them. The number of ways to arrange 2 letters is given by the factorial of the number of letters to arrange:
\[ \text{Number of arrangements of 2 letters} = 2! = 2 \times 1 = 2 \]

4. **Calculate the Total Permutations**:
To find the total number of permutations of the selected letters, we multiply the number of ways to choose the letters by the number of arrangements:
\[ \text{Total permutations} = \binom{4}{2} \times 2! = 6 \times 2 = 12 \]

5. **Final Answer**:
Therefore, the total number of permutations of the set of four letters E, O, S, P taken two at a time is **12**.