If `""^(n)P_(3)`= 9240 , then find the value of n.

Options

A. 20
B. 21
C. 22
D. 23

Answer

To solve the problem, we need to find the value of \( n \) given that \( nP3 = 9240 \). We will use the formula for permutations to solve this.

### Step-by-Step Solution:

1. **Understand the formula for permutations**:
The formula for permutations is given by:
\[ nP_r = \frac{n!}{(n-r)!} \]
Here, we have \( r = 3 \), so:
\[ nP_3 = \frac{n!}{(n-3)!} \]

2. **Set up the equation**:
According to the problem, we have:
\[ nP_3 = 9240 \]
Therefore, we can write:
\[ \frac{n!}{(n-3)!} = 9240 \]

3. **Simplify the equation**:
The factorial \( n! \) can be expressed as:
\[ n! = n \times (n-1) \times (n-2) \times (n-3)! \]
Substituting this into our equation gives:
\[ \frac{n \times (n-1) \times (n-2) \times (n-3)!}{(n-3)!} = 9240 \]
The \( (n-3)! \) terms cancel out:
\[ n \times (n-1) \times (n-2) = 9240 \]

4. **Finding \( n \)**:
Now we need to find \( n \) such that:
\[ n(n-1)(n-2) = 9240 \]
We can start testing values for \( n \).

5. **Testing values**:
Let's try \( n = 22 \):
\[ 22 \times 21 \times 20 \]
Calculate:
\[ 22 \times 21 = 462 \]
Then:
\[ 462 \times 20 = 9240 \]
This is correct.

6. **Conclusion**:
Therefore, the value of \( n \) is:
\[ n = 22 \]