Evaluate the following products:
(a) $ \Pi _{j=1}^{6}\left (2j \right ) $
(b) $ \Pi _{j=1}^{6} \left ( \frac{j}{2} \right ) $
(b) $ \Pi _{j=1}^{3} \Pi_{i=1}^{5}\left ( \frac{i}{j} \right ) $
(a) We have
$$ \begin{align} \Pi _{j=1}^{6}\left (2j \right ) & = (2 \times 1) \times (2 \times 2) \times (2 \times 3) \times (2 \times 4) \times (2 \times 5) \times (2 \times 6) \\ \\ & = 2^6 \times 6! \\ \\ & = 46080 \end{align}$$
(b) We have
$$ \begin{align} \Pi _{j=1}^{6}\left ( \frac{j}{2} \right ) & = \frac{1}{2} \times \frac{2}{2} \times \frac{3}{2} \times \frac{4}{2} \times \frac{5}{2} \times \frac{6}{2}\\ \\ & = \frac{6!}{2^6} \\ \\ & = \frac{45}{4} \end{align}$$
(c) We have
$$ \begin{align} \Pi _{j=1}^{3} \Pi_{i=1}^{5}\left ( \frac{i}{j} \right ) & = \left ( \frac{1}{1} \times \frac{2}{1} \times \frac{3}{1} \times \frac{4}{1} \times \frac{5}{1} \right) \times \left ( \frac{1}{2} \times \frac{2}{2} \times \frac{3}{2} \times \frac{4}{2} \times \frac{5}{2} \right) \times \left ( \frac{1}{3} \times \frac{2}{3} \times \frac{3}{3} \times \frac{4}{3} \times \frac{5}{3} \right) \\ \\ & = \frac{1728000}{7776} \\ \\ & = \frac{2000}{9} \end{align}$$