Question 24
Let $n$ be a positive integer. Show that the numbers
$$
\left(\begin{array}{c}
2^{n}-1 \\
0
\end{array}\right), \quad\left(\begin{array}{c}
2^{n}-1 \\
1
\end{array}\right), \quad\left(\begin{array}{c}
2^{n}-1 \\
2
\end{array}\right), \quad \ldots, \quad\left(\begin{array}{c}
2^{n}-1 \\
2^{n-1}-1
\end{array}\right)
$$
are congruent modulo $2^{n}$ to $1,3,5, \ldots, 2^{n}-1$ in some order.
$$
\left(\begin{array}{c}
2^{n}-1 \\
0
\end{array}\right), \quad\left(\begin{array}{c}
2^{n}-1 \\
1
\end{array}\right), \quad\left(\begin{array}{c}
2^{n}-1 \\
2
\end{array}\right), \quad \ldots, \quad\left(\begin{array}{c}
2^{n}-1 \\
2^{n-1}-1
\end{array}\right)
$$
are congruent modulo $2^{n}$ to $1,3,5, \ldots, 2^{n}-1$ in some order.