Binomial Coefficient is odd condition in binary system

Question

Here's a question that I encountered in Niven Zuckerman's Introduction to Theory of Numbers book (Problem 16,2.2) The question from the book part 1

Part 2 of the question

I showed the part where all coefficients of the polynomial are even. But I was stuck when I tried the ${n}\choose{k}$ is odd part. I tried by expanding n and k in binary and then computing the binomial coefficient but it is a dead end. I also couldn't understand and solve the binary weight part. Any ideas or suggestions will be very helpful. Thanks in advance.