Help me with this factorail question, please?

Question

Help me with this factorial, please?

Simplify the following expressions in the terms of factorials and powers of two: 2x4x6x....x2008, 1x3x5x....2007.

Could you please show me how you got the answer by work?
Thanks. <3

Answer 1

2x4x6x...x2008
= 2(1x2x3x...x1004)
= 2 * 1004!

1x3x5x...x2007
= 1x2x3x4x...x2008 / 2x4x6x8...x2008
= 2008! / 2(1004!)

Answer 2

the first one:
2 x 4 x 6 x 8 x ... x 2008
= 2x1 x 2x2 x 2x3 x 2x4 x ... x 2x1004 (group the x2)
= 2x2x2x2x...x2 x 1x2x3x4x...x1004
=2^1004 x 1004!

the second one is tougher:
2008!
= 1 x 2 x 3 x 4 x 5 x 6 x ... x 2007 x 2008
= 1x3x5x...x2007 x 2x4x6x...2008
so:
1x3x5x...x2007
= 2008! / 2x4x6x...x2008
= 2008! / (2^1004 x 1004!)

(2x4x6x...x2008=2^1004 x 1004! as shown before)

i hope that was clear enough.

P.S.: @Mathematica:
I'm afraid you're wrong
because 2(1x2x3x...x1004)
is not equal to 2x4x6x...x2008
but to 2x2x3x...x1004

2(1x2x3)=2x6=12
2x4x6=48 (not 12)

Answer 3

2•4•6•...•2008 =
(1•2)(2•2)(3•2)...(1004•2)
left hand factor gives you 1004!
right hand factor gives you 2^1004
so
2•4•6•...•2008 = (1004!)(2^1004)

1x3x5x...x2007 =

1x2x3x4x5x....x2006x2007
----------------------------------- =
2x4x6x....x2006

2007! / [(1003!)(2^1003)]