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 Letters**:
We have a set of four letters: E, O, S, P.

2. **Determine the Selection Size**:
We need to select 2 letters at a time from these 4 letters.

3. **Calculate the Combinations**:
The number of ways to choose 2 letters from 4 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 4 \) and \( r = 2 \):
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]
So, there are 6 ways to choose 2 letters from the 4 letters.

4. **Calculate the Permutations of the Selected Letters**:
For each selection of 2 letters, we can arrange them in different ways. The number of arrangements (permutations) of 2 letters is given by \( 2! \):
\[ 2! = 2 \times 1 = 2 \]

5. **Calculate the Total Permutations**:
Now, we multiply the number of combinations by the number of permutations:
\[ \text{Total permutations} = \text{Number of combinations} \times \text{Permutations of selected letters} = 6 \times 2 = 12 \]

6. **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**.