If `""^(56)P_(r +6) :""^(54)P_(r+3)= 30800 :1` find r.

Options

A. 14
B. 20
C. 41
D. 21

Answer

To solve the problem, we start with the equation given in the question:

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

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

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

Using this, we can express the left-hand side:

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

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

### Step 2: Substitute the permutation formulas into the equation
Now, substituting these into our equation gives:

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

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

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

We can simplify \( \frac{56!}{54!} \) to \( 56 \times 55 \):

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

### Step 4: Further simplify
Now, we can express \( \frac{(51 - r)!}{(50 - r)!} \) as \( (51 - r) \):

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

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

\[ 56 \times 55 = 3080 \]

### Step 6: Substitute back into the equation
Now we have:

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

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

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

### Step 8: Solve for \( r \)
Now, we can solve for \( r \):

\[ r = 51 - 10 = 41 \]

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

\[ \boxed{41} \]