If `""^10 P_r=5040 ,` find the value of `r.`

Options

A. 5
B. 4
C. 6
D. None of these

Answer

To solve the problem where \( ^{10}P_r = 5040 \), we will follow these steps:

### Step 1: Understand the formula for permutations
The formula for permutations is given by:
\[ ^{n}P_r = \frac{n!}{(n-r)!} \]
In this case, \( n = 10 \), so we have:
\[ ^{10}P_r = \frac{10!}{(10-r)!} \]

### Step 2: Set up the equation
According to the problem, we have:
\[ \frac{10!}{(10-r)!} = 5040 \]

### Step 3: Calculate \( 10! \)
We know that:
\[ 10! = 3628800 \]
So we can rewrite the equation as:
\[ \frac{3628800}{(10-r)!} = 5040 \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3628800 = 5040 \times (10-r)! \]

### Step 5: Divide both sides by 5040
Now, we can divide both sides by 5040 to isolate \( (10-r)! \):
\[ (10-r)! = \frac{3628800}{5040} \]

### Step 6: Calculate the right-hand side
Calculating the right-hand side:
\[ \frac{3628800}{5040} = 720 \]
So, we have:
\[ (10-r)! = 720 \]

### Step 7: Find the factorial value
We know that:
\[ 6! = 720 \]
Thus, we can equate:
\[ 10 - r = 6 \]

### Step 8: Solve for \( r \)
Now, solving for \( r \):
\[ r = 10 - 6 = 4 \]

### Conclusion
The value of \( r \) is:
\[ \boxed{4} \]