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)/(11)=(b)/(22)=(c)/(21)`
C. ` (a)/(10) =(b)/(11)=(c)/(21)`
D. ` (a)/(10) =(b)/(11)= (c)/(42)`

Answer

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-by-Step Solution:

1. **Identify the Maximum Values**:
The maximum value of \( \binom{n}{k} \) occurs when \( k = \frac{n}{2} \) (or the nearest integer). Therefore:
- For \( \binom{19}{p} \), the maximum occurs at \( p = 9 \) (since \( \frac{19}{2} = 9.5 \)).
- For \( \binom{20}{q} \), the maximum occurs at \( q = 10 \) (since \( \frac{20}{2} = 10 \)).
- For \( \binom{21}{r} \), the maximum occurs at \( r = 10 \) (since \( \frac{21}{2} = 10.5 \), we take the nearest integer).

2. **Express the Maximum Values**:
We denote:
- \( A = \binom{19}{9} \)
- \( B = \binom{20}{10} \)
- \( C = \binom{21}{10} \)

3. **Use the Formula for Binomial Coefficients**:
The binomial coefficient is given by:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Thus, we can write:
- \( A = \frac{19!}{9!10!} \)
- \( B = \frac{20!}{10!10!} \)
- \( C = \frac{21!}{10!11!} \)

4. **Set Up the Ratios**:
We will set the ratios:
\[ \frac{A}{\binom{19}{9}} = \frac{B}{\binom{20}{10}} = \frac{C}{\binom{21}{10}} = k \]
where \( k \) is a constant.

5. **Simplify Each Ratio**:
- For \( A \):
\[ A = \frac{19!}{9!10!} \]
- For \( B \):
\[ B = \frac{20!}{10!10!} = \frac{20 \cdot 19!}{10! \cdot 10!} \]
- For \( C \):
\[ C = \frac{21!}{10!11!} = \frac{21 \cdot 20!}{10! \cdot 11!} = \frac{21 \cdot 20 \cdot 19!}{10! \cdot 11 \cdot 10!} \]

6. **Set Up the Equalities**:
From the ratios:
\[ \frac{A}{\binom{19}{9}} = \frac{B}{\binom{20}{10}} = \frac{C}{\binom{21}{10}} = k \]
This leads to:
\[ \frac{A}{\frac{19!}{9!10!}} = \frac{B}{\frac{20 \cdot 19!}{10! \cdot 10!}} = \frac{C}{\frac{21 \cdot 20 \cdot 19!}{10! \cdot 11 \cdot 10!}} \]

7. **Final Relations**:
After simplifying, we find:
\[ \frac{A}{1} = \frac{B}{2} = \frac{C}{42} \]
Thus, we can express the relationships as:
\[ A : B : C = 1 : 2 : 42 \]

### Conclusion:
The final relationship between \( a \), \( b \), and \( c \) is:
\[ A : B : C = 1 : 2 : 42 \]