Step-by-step Solution:

Step 1: Understanding the Binomial Coefficients \[\] The binomial coefficient is given by: \[ ^mC_r = \frac{m!}{r!(m-r)!} \] The greatest value of \( ^mC_r \) occurs at \( r \) close to \( \frac{m}{2} \). \[\] This is because binomial coefficients are symmetric and attain their maximum at the middle value. \[\] Step 2: Greatest Value of \( ^{2n}C_r \) \[\] For \( ^{2n}C_r \), the maximum value occurs at: \[ r = \frac{2n}{2} = n \] Thus, the maximum value is: \[ a = ^{2n}C_n = \frac{(2n)!}{n!n!} \] Step 3: Greatest Value of \( ^{2n-1}C_r \) \[\] For \( ^{2n-1}C_r \), the maximum value occurs at: \[ r = \frac{2n-1}{2} = n-1 \quad \text{or} \quad n \] Since \( ^mC_r = ^mC_{m-r} \), both values are equal, and we take one: \[ b = ^{2n-1}C_{n-1} = \frac{(2n-1)!}{(n-1)!(n)!} \] Step 4: Finding the Relationship Between \( a \) and \( b \) \[\] By simplifying, \[ \frac{a}{b} = \frac{^{2n}C_n}{^{2n-1}C_{n-1}} \] Expanding the factorial expressions: \[ \frac{\frac{(2n)!}{n!n!}}{\frac{(2n-1)!}{(n-1)!n!}} \] \[ = \frac{(2n)!}{(2n-1)!} \times \frac{(n-1)!}{n!} \] \[ = \frac{(2n)(2n-1)!}{(2n-1)!} \times \frac{(n-1)!}{n(n-1)!} \] \[ = \frac{2n}{n} = 2 \] Conclusion \[ a = 2b \]