Question 4
Prove that for every positive integer $t$ there is a unique permutation $a_0,a_1,\ldots,a_{t-1}$ of $0,1,\ldots,t-1$ such that, for every $0\le i\le t-1$, the binomial coefficient
$$\binom{t+i}{2a_i}$$
is odd and $2a_i\ne t+i$.
$$\binom{t+i}{2a_i}$$
is odd and $2a_i\ne t+i$.