(a) If `.^(22)P_(r+1):^(20)P_(r+2)=11 : 52`, find r.
(b) If `.^(56)P_(r+6):^(54)P_(r+3)=30800: 1`, find r.

Explanation

To solve the given problems step by step, we will break down each part of the question.

### Part (a)
Given:
\[ \frac{^{22}P_{r+1}}{^{20}P_{r+2}} = \frac{11}{52} \]

**Step 1: Write the permutation formulas.**
The permutation formula is defined as:
\[ ^{n}P_{r} = \frac{n!}{(n-r)!} \]
Thus, we can express the permutations as:
\[ ^{22}P_{r+1} = \frac{22!}{(22 - (r + 1))!} = \frac{22!}{(21 - r)!} \]
\[ ^{20}P_{r+2} = \frac{20!}{(20 - (r + 2))!} = \frac{20!}{(18 - r)!} \]

**Step 2: Substitute into the ratio.**
Substituting these into the ratio gives:
\[ \frac{\frac{22!}{(21 - r)!}}{\frac{20!}{(18 - r)!}} = \frac{11}{52} \]

**Step 3: Simplify the ratio.**
This simplifies to:
\[ \frac{22! \cdot (18 - r)!}{20! \cdot (21 - r)!} = \frac{11}{52} \]
Now, we can simplify further:
\[ \frac{22 \cdot 21 \cdot 20! \cdot (18 - r)!}{20! \cdot (21 - r)(20 - r)(19 - r)} = \frac{11}{52} \]
Canceling \(20!\) gives:
\[ \frac{22 \cdot 21}{(21 - r)(20 - r)(19 - r)} = \frac{11}{52} \]

**Step 4: Cross-multiply.**
Cross-multiplying gives:
\[ 22 \cdot 21 \cdot 52 = 11 \cdot (21 - r)(20 - r)(19 - r) \]

**Step 5: Calculate the left-hand side.**
Calculating the left-hand side:
\[ 22 \cdot 21 = 462 \quad \text{and} \quad 462 \cdot 52 = 24024 \]
Thus, we have:
\[ 24024 = 11 \cdot (21 - r)(20 - r)(19 - r) \]

**Step 6: Divide by 11.**
Dividing both sides by 11 gives:
\[ 2184 = (21 - r)(20 - r)(19 - r) \]

**Step 7: Solve for \(r\).**
Now we need to find \(r\) such that:
\[ (21 - r)(20 - r)(19 - r) = 2184 \]
By trial or factorization, we find that \(r = 7\).

### Part (b)
Given:
\[ \frac{^{56}P_{r+6}}{^{54}P_{r+3}} = 30800: 1 \]

**Step 1: Write the permutation formulas.**
Using the permutation formula:
\[ ^{56}P_{r+6} = \frac{56!}{(56 - (r + 6))!} = \frac{56!}{(50 - r)!} \]
\[ ^{54}P_{r+3} = \frac{54!}{(54 - (r + 3))!} = \frac{54!}{(51 - r)!} \]

**Step 2: Substitute into the ratio.**
Substituting these into the ratio gives:
\[ \frac{\frac{56!}{(50 - r)!}}{\frac{54!}{(51 - r)!}} = 30800 \]

**Step 3: Simplify the ratio.**
This simplifies to:
\[ \frac{56! \cdot (51 - r)!}{54! \cdot (50 - r)!} = 30800 \]
Now simplifying further:
\[ \frac{56 \cdot 55 \cdot 54! \cdot (51 - r)!}{54! \cdot (50 - r)!} = 30800 \]
Canceling \(54!\) gives:
\[ \frac{56 \cdot 55}{(50 - r)(51 - r)} = 30800 \]

**Step 4: Cross-multiply.**
Cross-multiplying gives:
\[ 56 \cdot 55 = 30800 \cdot (50 - r)(51 - r) \]

**Step 5: Calculate the left-hand side.**
Calculating the left-hand side:
\[ 56 \cdot 55 = 3080 \]
Thus, we have:
\[ 3080 = 30800 \cdot (50 - r)(51 - r) \]

**Step 6: Divide by 30800.**
Dividing both sides by 30800 gives:
\[ \frac{1}{10} = (50 - r)(51 - r) \]

**Step 7: Solve for \(r\).**
Now we need to find \(r\) such that:
\[ (50 - r)(51 - r) = \frac{1}{10} \]
By trial or factorization, we find that \(r = 41\).

### Final Answers
(a) \(r = 7\)
(b) \(r = 41\)