IF `""^56 P_(r+6) : "^(54) P_(r+3)=30800:1` then r=

Options

A. 40
B. 41
C. 42
D. 43

Answer

To solve the problem, we need to work with the given equation involving permutations. The equation is:

\[ \frac{56P_{(r+6)}}{54P_{(r+3)}} = \frac{30800}{1} \]

### Step 1: Write the permutation formula
The permutation formula is given by:

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

Using this, we can express \(56P_{(r+6)}\) and \(54P_{(r+3)}\):

\[ 56P_{(r+6)} = \frac{56!}{(56 - (r + 6))!} = \frac{56!}{(50 - r)!} \]

\[ 54P_{(r+3)} = \frac{54!}{(54 - (r + 3))!} = \frac{54!}{(51 - r)!} \]

### Step 2: Substitute into the equation
Now substituting these into our equation, we have:

\[ \frac{\frac{56!}{(50 - r)!}}{\frac{54!}{(51 - r)!}} = 30800 \]

### Step 3: Simplify the left side
This simplifies to:

\[ \frac{56! \cdot (51 - r)!}{54! \cdot (50 - r)!} \]

We know that \(56! = 56 \times 55 \times 54!\), so we can substitute:

\[ \frac{56 \times 55 \times 54! \cdot (51 - r)!}{54! \cdot (50 - r)!} \]

The \(54!\) cancels out:

\[ \frac{56 \times 55 \cdot (51 - r)!}{(50 - r)!} = 30800 \]

### Step 4: Further simplify
Now we can express \((51 - r)!\) in terms of \((50 - r)!\):

\[ (51 - r)! = (51 - r) \cdot (50 - r)! \]

Substituting this into our equation gives:

\[ \frac{56 \times 55 \cdot (51 - r) \cdot (50 - r)!}{(50 - r)!} = 30800 \]

The \((50 - r)!\) cancels out again:

\[ 56 \times 55 \times (51 - r) = 30800 \]

### Step 5: Calculate \(56 \times 55\)
Calculating \(56 \times 55\):

\[ 56 \times 55 = 3080 \]

### Step 6: Set up the equation
Now we have:

\[ 3080 \times (51 - r) = 30800 \]

### Step 7: Solve for \(51 - r\)
Dividing both sides by 3080 gives:

\[ 51 - r = \frac{30800}{3080} = 10 \]

### Step 8: Solve for \(r\)
Now, solving for \(r\):

\[ r = 51 - 10 = 41 \]

Thus, the value of \(r\) is:

\[ \boxed{41} \]