How many zeroes at the end of binary representation of 50!

Question

The answers don't make sense to me. The binary number of 50 is 110010. Do I overlook something?

Answer 1

The number of zeroes at the end of $n$ in base $k$ notation is the highest power of $k$ that divides $n$. This is very elementary proof.

Now, the highest power of $2$ that divides $50!$ can be calculated as:
$\sum\limits_{i = 1}^\infty\lfloor\frac{50}{2^i}\rfloor = \lfloor\frac{50}{2}\rfloor + \lfloor\frac{50}{4}\rfloor + \cdots = 25 + 16 + 6 + 3 + 1 = 47$, because the terms $\lfloor\frac{50}{2^6}\rfloor$ and beyond will be $0$.

Answer 2

Or you can write a simple Python programme to calculate it:

>>> from math import *
>>> bin(factorial(50))
    ....
>>> len('copy and paste here the zero-s at the end')
    47