Let N=1!⋅2!⋅3!⋅4!…9!⋅10!. Let 2^k be the largest power of 2 that divides N. What is the value of k?

Answer 1

Count how many 2's there are in each of the factorials,
1! = none
2! = 1
3! = 1
4! = 2*4 => 3
5! = 2*4 => 3
6! = 2*4*2 => 4
7! ...
10!=2*4*8*2 => 8
Add all the numbers on the right to get N.

Answer 2

38