Question 23
Given a polynomial $P(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{k} x^{k}$ with integer coefficients, we denote by $w(P)$ the number of odd coefficients of $P$. Let $Q_{i}(x)=(1+x)^{i}$, for $i=0,1, \ldots$ to show that if $i_{1}, i_{2}, \ldots, i_{n}$ are such as $0 \leq i_{1}\lt i_{2}\lt \cdots\lt i_{n}$, then $$
w\left(Q_{i_{1}}+Q_{i_{2}}+\cdots+Q_{i_{n}}\right) \geq w\left(Q_{i_{1}}\right)
$$
w\left(Q_{i_{1}}+Q_{i_{2}}+\cdots+Q_{i_{n}}\right) \geq w\left(Q_{i_{1}}\right)
$$