(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

Let's solve the given problems step by step.

### Part (a)

We are given the ratio:

\[ \frac{^{22}P_{r+1}}{^{20}P_{r+2}} = \frac{11}{52} \]

Using the formula for permutations, we have:

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

So, we can write:

\[ ^{22}P_{r+1} = \frac{22!}{(22 - (r + 1))!} = \frac{22!}{(21 - r)!} \]

And,

\[ ^{20}P_{r+2} = \frac{20!}{(20 - (r + 2))!} = \frac{20!}{(18 - r)!} \]

Now substituting these into the ratio:

\[ \frac{\frac{22!}{(21 - r)!}}{\frac{20!}{(18 - r)!}} = \frac{11}{52} \]

This simplifies to:

\[ \frac{22! \cdot (18 - r)!}{20! \cdot (21 - r)!} = \frac{11}{52} \]

Now, we can simplify \( \frac{22!}{20!} = 22 \times 21 \):

\[ \frac{22 \times 21 \cdot (18 - r)!}{(21 - r)(20 - r)(19 - r)(18 - r)!} = \frac{11}{52} \]

Canceling \( (18 - r)! \):

\[ \frac{22 \times 21}{(21 - r)(20 - r)(19 - r)} = \frac{11}{52} \]

Cross-multiplying gives:

\[ 52 \cdot 22 \cdot 21 = 11 \cdot (21 - r)(20 - r)(19 - r) \]

Calculating \( 52 \cdot 22 \cdot 21 \):

\[ 52 \cdot 22 = 1144 \quad \text{and} \quad 1144 \cdot 21 = 24024 \]

So we have:

\[ 24024 = 11 \cdot (21 - r)(20 - r)(19 - r) \]

Dividing both sides by 11:

\[ 2184 = (21 - r)(20 - r)(19 - r) \]

Now we can solve for \( r \). We can test values for \( r \) to find a suitable integer solution.

After testing, we find:

If \( r = 7 \):

\[ (21 - 7)(20 - 7)(19 - 7) = 14 \cdot 13 \cdot 12 = 2184 \]

Thus, \( r = 7 \) is a solution.

### Part (b)

We are given the ratio:

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

Using the permutation formula:

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

And,

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

Substituting into the ratio gives:

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

This simplifies to:

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

Now, simplifying \( \frac{56!}{54!} = 56 \times 55 \):

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

This can be rewritten as:

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

Calculating \( 56 \times 55 \):

\[ 56 \times 55 = 3080 \]

So we have:

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

Dividing both sides by 3080:

\[ 51 - r = 10 \]

Thus:

\[ r = 41 \]

### Final Answers

(a) \( r = 7 \)
(b) \( r = 41 \)