Find the value of `""^(5)P_(2)`.

Options

A. 15
B. 18
C. 20
D. 122

Answer

To find the value of \( ^{5}P_{2} \), we can use the formula for permutations, which is given by:

\[ ^{n}P_{r} = \frac{n!}{(n-r)!} \]

In this case, \( n = 5 \) and \( r = 2 \).

### Step 1: Substitute the values into the formula
We start by substituting \( n \) and \( r \) into the formula:

\[ ^{5}P_{2} = \frac{5!}{(5-2)!} \]

### Step 2: Simplify the expression
Now, simplify the expression:

\[ ^{5}P_{2} = \frac{5!}{3!} \]

### Step 3: Calculate the factorials
Next, we calculate the factorials:

- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

### Step 4: Substitute the factorial values back into the equation
Now, we substitute the values of the factorials back into the equation:

\[ ^{5}P_{2} = \frac{120}{6} \]

### Step 5: Perform the division
Finally, we perform the division:

\[ ^{5}P_{2} = 20 \]

Thus, the value of \( ^{5}P_{2} \) is \( 20 \).

---