If maximum value of `.^19 C_p` is a, `.^20 C_q` is b, `.^21 C_r` is c`, then relation between a, b, c is

Options

A. `a/11 = b/22 = c/42`
B. `a/22 = b/11 = c/42`
C. `a/22 = b/42 = c/11`
D. `a/21 = b/11 = c/22`

Explanation

To solve the problem, we need to find the relationship between the maximum values of the binomial coefficients \( \binom{19}{p} \), \( \binom{20}{q} \), and \( \binom{21}{r} \).

### Step 1: Identify the maximum values of the binomial coefficients
The maximum value of \( \binom{n}{k} \) occurs at \( k = \lfloor \frac{n}{2} \rfloor \).

- For \( \binom{19}{p} \), the maximum occurs at \( p = 9 \) (since \( \lfloor \frac{19}{2} \rfloor = 9 \)).
- For \( \binom{20}{q} \), the maximum occurs at \( q = 10 \) (since \( \lfloor \frac{20}{2} \rfloor = 10 \)).
- For \( \binom{21}{r} \), the maximum occurs at \( r = 10 \) (since \( \lfloor \frac{21}{2} \rfloor = 10 \)).

Thus, we have:
- \( a = \binom{19}{9} \)
- \( b = \binom{20}{10} \)
- \( c = \binom{21}{10} \)

### Step 2: Express the binomial coefficients
Now we can express these coefficients using factorials:
- \( a = \binom{19}{9} = \frac{19!}{9! \cdot 10!} \)
- \( b = \binom{20}{10} = \frac{20!}{10! \cdot 10!} \)
- \( c = \binom{21}{10} = \frac{21!}{10! \cdot 11!} \)

### Step 3: Set up the ratios
We need to find the ratios \( \frac{a}{\binom{19}{9}} \), \( \frac{b}{\binom{20}{10}} \), and \( \frac{c}{\binom{21}{10}} \):
\[ \frac{a}{\binom{19}{9}} = 1, \quad \frac{b}{\binom{20}{10}} = 1, \quad \frac{c}{\binom{21}{10}} = 1 \]

### Step 4: Find the relationship
Now we can relate \( a \), \( b \), and \( c \) using their expressions:
\[ \frac{a}{b} = \frac{\binom{19}{9}}{\binom{20}{10}} = \frac{\frac{19!}{9! \cdot 10!}}{\frac{20!}{10! \cdot 10!}} = \frac{19! \cdot 10!}{20! \cdot 9!} = \frac{10}{20} = \frac{1}{2} \]
Thus, \( a = \frac{1}{2}b \).

Next, we find \( \frac{b}{c} \):
\[ \frac{b}{c} = \frac{\binom{20}{10}}{\binom{21}{10}} = \frac{\frac{20!}{10! \cdot 10!}}{\frac{21!}{10! \cdot 11!}} = \frac{20! \cdot 11!}{21! \cdot 10!} = \frac{11}{21} \]
Thus, \( b = \frac{11}{21}c \).

### Conclusion
From the ratios we derived:
1. \( a = \frac{1}{2}b \)
2. \( b = \frac{11}{21}c \)

We can express the relationships as:
\[ a : b : c = 1 : 2 : \frac{21}{11} \]

### Final Relation
The final relation between \( a \), \( b \), and \( c \) can be summarized as:
\[ a : b : c = 1 : 2 : \frac{21}{11} \]