Decide if the binomial coefficient $\displaystyle {100 \choose 68} $ is even or odd.
We know (and if not prove it!) that the maximal power of a prime $\displaystyle p$ dividing $\displaystyle n!$ is given by $\displaystyle \sum^\infty_{k=1}\left[n/p^k\right]$ , where $\displaystyle [x]$ is the floor function = the greatest integer that is less than or equal $\displaystyle x$.
Note that the above sum is finite since $\displaystyle \left[n/p^r\right]=0\,\,\,for\,\,\,p^r>n$
So the maximal power of 2 dividing 100! is $\displaystyle 50+25+12+6+3+1=97$ , and the max. power of 2 dividing 32! is $\displaystyle 16+8+4+2+1=31$ , and the one
dividing 68 is $\displaystyle 34+17+8+4+2+1=66$ , thus the number is odd.
Tonio