Step 1: Write each exponent as twice a whole number, \(a=2p,\ b=2q,\ c=2r,\ d=2s\), since every exponent must be even. Substituting \(N = 2^{11}3^{11}5^9 7^{10}\) and \(420=2^2\cdot3\cdot5\cdot7\) turns "even and at least 420's exponent" into simple bounds on \(p,q,r,s\).
Step 2: For \(2^{2p}\): \(2p\ge2\) and \(2p\le11\), so \(1\le p\le5\) (5 values). For \(3^{2q}\): \(2q\ge1\Rightarrow q\ge1\), and \(2q\le11\Rightarrow q\le5\) (5 values). For \(5^{2r}\): \(r\ge1,\ 2r\le9\Rightarrow r\le4\) (4 values). For \(7^{2s}\): \(s\ge1,\ 2s\le10\Rightarrow s\le5\) (5 values).
Step 3: Multiply the counts:
\[ 5\times5\times4\times5 = \boxed{500} \]
Final Answer: 500.