Question 10
Let $P$ be the number obtained by multiplying the factorials of the first 2008 positive integers:
$$
P = (1!)(2!)(3!) \ldots (2007!)(2008!)
$$
Determine whether it is possible to cancel one of these factorials so that the product of the 2007 remaining factorials is a perfect square.
**CLARIFICATION:** The factorial of a positive integer is the product of all integers from 1 up to that number. For example, $5!=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$; $12!=1 \cdot 2 \cdot 3 \cdot ... \cdot 12 = 479001600$. An integer is called a perfect square if it is the square of an integer. For example, 16 and 10000 are perfect squares, because $16 = 4^2$ and $10000 = 100^2$.
$$
P = (1!)(2!)(3!) \ldots (2007!)(2008!)
$$
Determine whether it is possible to cancel one of these factorials so that the product of the 2007 remaining factorials is a perfect square.
**CLARIFICATION:** The factorial of a positive integer is the product of all integers from 1 up to that number. For example, $5!=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$; $12!=1 \cdot 2 \cdot 3 \cdot ... \cdot 12 = 479001600$. An integer is called a perfect square if it is the square of an integer. For example, 16 and 10000 are perfect squares, because $16 = 4^2$ and $10000 = 100^2$.