`"^n P_r=`

Options

A. `""^(n-1)P_r+r. "^(n-1) P_(r-1)`
B. `(n!)/(r!(n-r)!)`
C. `r.""^(n-1)P_r- "^((n-1))P_(r-1)`
D. `""^(n-1) P_r+""^((n-1))P_((r-1))`

Answer

To solve the problem of finding the value of \( nP_r \), we can use the formula for permutations and the properties of combinations. Here’s a step-by-step breakdown of the solution:

### Step-by-Step Solution:

1. **Understanding the Formula for Permutations**:
The formula for permutations is given by:
\[ nP_r = \frac{n!}{(n-r)!} \]

2. **Using the Recursive Relation**:
We can express \( nP_r \) in terms of \( n-1 \):
\[ nP_r = n \cdot (n-1)P_{r-1} \]
This means that the number of ways to arrange \( r \) items from \( n \) items can be thought of as choosing one item (which can be done in \( n \) ways) and then arranging the remaining \( r-1 \) items from the remaining \( n-1 \) items.

3. **Using the Combination Formula**:
We can also express \( nP_r \) in terms of combinations:
\[ nP_r = nC_r \cdot r! \]
Where \( nC_r \) is the number of ways to choose \( r \) items from \( n \) items, and \( r! \) is the number of ways to arrange those \( r \) items.

4. **Identifying the Correct Option**:
The options given in the question can be analyzed:
- The first option is \( n-1P_r + r \cdot n-1P_{r-1} \).
- The second option is \( \frac{n!}{r!(n-r)!} \), which is the formula for combinations \( nC_r \).
- The third option is \( r \cdot n-1P_{r-1} \).
- The fourth option is \( n-1P_r + n-1P_{r-1} \).

5. **Simplifying the First Option**:
Let's simplify the first option:
\[ n-1P_r + r \cdot n-1P_{r-1} \]
Using the formula for permutations:
\[ n-1P_r = \frac{(n-1)!}{(n-1-r)!} \]
and
\[ n-1P_{r-1} = \frac{(n-1)!}{(n-1-(r-1))!} = \frac{(n-1)!}{(n-r)!} \]
Thus,
\[ n-1P_r + r \cdot n-1P_{r-1} = \frac{(n-1)!}{(n-1-r)!} + r \cdot \frac{(n-1)!}{(n-r)!} \]
Factoring out \( (n-1)! \):
\[ = (n-1)! \left( \frac{1}{(n-1-r)!} + r \cdot \frac{1}{(n-r)!} \right) \]
This simplifies to:
\[ = (n-1)! \cdot \frac{(n-r) + r}{(n-r)!} = \frac{n!}{(n-r)!} \]
Hence, we confirm that:
\[ nP_r = n-1P_r + r \cdot n-1P_{r-1} \]

### Conclusion:
The value of \( nP_r \) can be expressed as:
\[ nP_r = n-1P_r + r \cdot n-1P_{r-1} \]
Thus, the correct option is the first one.