粗心的小欣在計算1!×2!×3!×......×2016! 時漏算了一個階乘項,誤得結果為一平方數,問她漏算了那一項?

1! × 2! × 3! ×......× 2016!
= 2^2015 × 3^2014 × 4^2013 × 5^2012 × ... × 2015² × 2016
= (3^1007 × 5^1006 × ... × 2015)² × 2^2015 × 4^2013 × ... × 2016
= (3^1007 × 5^1006 × ... × 2015)² × (2^2014 × 4^2012 × ... × 1) × (2 × 4 × ... × 2016)
= (3^1007 × 5^1006 × ... × 2015)² × (2^1007 × 4^1006 × ... × 1)² × (2^1008) × (1 × 2 × ... × 1008)
= (3^1007 × 5^1006 × ... × 2015)² × (2^1007 × 4^1006 × ... × 1)² × (2^504)² × 1008!
她漏算了的一項是 1008! 。