If ` ""^(15)P_(r-1): ""^(16)P_(r-2)=3:4`, find r.

Explanation

To solve the problem, we start with the given ratio of permutations:

\[ \frac{^{15}P_{r-1}}{^{16}P_{r-2}} = \frac{3}{4} \]

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

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

### Step 2: Apply the formula to the permutations
Using the formula, we can express \(^{15}P_{r-1}\) and \(^{16}P_{r-2}\):

\[ ^{15}P_{r-1} = \frac{15!}{(15 - (r-1))!} = \frac{15!}{(16 - r)!} \]

\[ ^{16}P_{r-2} = \frac{16!}{(16 - (r-2))!} = \frac{16!}{(18 - r)!} \]

### Step 3: Substitute into the ratio
Now, substitute these into the ratio:

\[ \frac{\frac{15!}{(16 - r)!}}{\frac{16!}{(18 - r)!}} = \frac{3}{4} \]

### Step 4: Simplify the ratio
This simplifies to:

\[ \frac{15! \cdot (18 - r)!}{16! \cdot (16 - r)!} = \frac{3}{4} \]

### Step 5: Rewrite \(16!\)
We know that \(16! = 16 \cdot 15!\), so we can rewrite the equation as:

\[ \frac{15! \cdot (18 - r)!}{16 \cdot 15! \cdot (16 - r)!} = \frac{3}{4} \]

### Step 6: Cancel \(15!\)
Cancel \(15!\) from both sides:

\[ \frac{(18 - r)!}{16 \cdot (16 - r)!} = \frac{3}{4} \]

### Step 7: Cross-multiply
Cross-multiply to eliminate the fraction:

\[ 4(18 - r)! = 48(16 - r)! \]

### Step 8: Rewrite \( (18 - r)! \)
We can express \( (18 - r)! \) in terms of \( (16 - r)! \):

\[ (18 - r)! = (18 - r)(17 - r)(16 - r)! \]

Substituting this back into the equation gives:

\[ 4(18 - r)(17 - r)(16 - r)! = 48(16 - r)! \]

### Step 9: Cancel \( (16 - r)! \)
Assuming \( (16 - r)! \neq 0 \), we can cancel it:

\[ 4(18 - r)(17 - r) = 48 \]

### Step 10: Simplify the equation
Dividing both sides by 4:

\[ (18 - r)(17 - r) = 12 \]

### Step 11: Expand and rearrange
Expanding the left side gives:

\[ 306 - 35r + r^2 = 12 \]

Rearranging this leads to:

\[ r^2 - 35r + 294 = 0 \]

### Step 12: Solve the quadratic equation
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1, b = -35, c = 294 \):

\[ b^2 - 4ac = 35^2 - 4 \cdot 1 \cdot 294 = 1225 - 1176 = 49 \]

Thus,

\[ r = \frac{35 \pm 7}{2} \]

Calculating the two possible values:

1. \( r = \frac{42}{2} = 21 \)
2. \( r = \frac{28}{2} = 14 \)

### Step 13: Determine the valid value of r
Since \( r \) must be less than or equal to 16 (as \( ^{15}P_{r-1} \) and \( ^{16}P_{r-2} \) are defined), we take:

\[ r = 14 \]

### Final Answer:
Thus, the value of \( r \) is:

\[ \boxed{14} \]