Question

Prove that: ⁴ⁿC_2n : ²ⁿC_n = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]².

Answer 1

\[\frac{{}^{4n} C_{2n}}{{}^{2n} C_n} = \frac{1 . 3 . 5 . . . \left( 4n - 1 \right)}{\left[ 1 . 3 . 5 . . . \left( 2n - 1 \right) \right]^2}\]
\[LHS = \frac{{}^{4n} C_{2n}}{{}^{2n} C_n}\]
\[ = \frac{\left( 4n \right)!}{\left( 2n \right)!\left( 2n \right)!} \times \frac{n!n!}{\left( 2n \right)!}\]
\[ = \frac{\left[ 4n \times \left( 4n - 1 \right) \times \left( 4n - 2 \right) \times \left( 4n - 3 \right) . . . . . . . . . . . . . . . . . 3 \times 2 \times 1 \right] \times \left( n! \right)^2}{\left[ 2n \times \left( 2n - 1 \right) \times \left( 2n - 2 \right) . . . . . . . 3 \times 2 \times \times 1 \right]^2 \left( 2n \right)!}\]
\[ = \frac{\left[ 1 \times 3 \times 5 . . . . . . . . \left( 4n - 1 \right) \right]\left[ 2 \times 4 \times 6 . . . . . . . . . . . . . . . 4n \right] \times \left( n! \right)^2}{\left[ 1 \times 3 \times 5 \times . . . . . . . . . \left( 2n - 1 \right) \right]^2 \left[ 2 \times 4 \times 6 \times . . . . . . . 2n \right]^2 \times \left( 2n \right)!}\]
\[ = \frac{\left[ 1 \times 3 \times 5 . . . . . . . . \left( 4n - 1 \right) \right] \times 2^{2n} \times \left[ 1 \times 2 \times 3 . . . . . . . . . . 2n \right] \left( n! \right)^2}{\left[ 1 \times 3 \times 5 \times . . . . . . . . . \left( 2n - 1 \right) \right]^2 \times 2^{2n} \left[ 1 \times 2 \times 3 \times . . . . . . . n \right]^2 \left( 2n \right)!}\]
\[ = \frac{\left[ 1 \times 3 \times 5 . . . . . . . . \left( 4n - 1 \right) \right]\left( 2n \right)! \left[ n! \right]^2}{\left[ 1 \times 3 \times 5 \times . . . . . . . . . \left( 2n - 1 \right) \right]^2 \left[ n! \right]^2 \left( 2n \right)!}\]
\[ = \frac{\left[ 1 \times 3 \times 5 . . . . . . . . \left( 4n - 1 \right) \right]}{\left[ 1 \times 3 \times 5 \times . . . . . . . . . \left( 2n - 1 \right) \right]^2} = RHS\]
\[\text{Hence, proved} .\]