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Exercises 23.4 Exercises
1.
\begin{equation*}
\sum_{k=0}^n \binom{n}{k}\, 2^k = 3^n \text{.}
\end{equation*}
2.
(a)
\begin{equation*}
\binom{n}{0} \;\;-\;\;
\binom{n}{1} \;\;+\;\;
\binom{n}{2} \;\;-\;\;
\binom{n}{3} \;\;+\;\;
\dotsb \;\;+\;\;
(-1)^n \binom{n}{n} \;\;=\;\;
0\text{.}
\end{equation*}
(b)
The equality from
Task a can be rearranged to yield
\begin{gather*}
\binom{n}{0} \;\;+\;\;
\binom{n}{2} \;\;+\;\;
\binom{n}{4} \;\;+\;\;
\dotsb \;\;+\;\;
\binom{n}{m_1}\\
\;\;=\;\;
\binom{n}{1} \;\;+\;\;
\binom{n}{3} \;\;+\;\;
\binom{n}{5} \;\;+\;\;
\dotsb \;\;+\;\;
\binom{n}{m_2}\text{,}
\end{gather*}
where
\begin{align*}
m_1 \amp = \begin{cases}
n, \amp n\text{ even}, \\
n-1, \amp n\text{ odd}, \\
\end{cases}
\amp
m_2 \amp = \begin{cases}
n-1, \amp n\text{ even}, \\
n, \amp n\text{ odd}. \\
\end{cases}
\end{align*}
What does this rearranged formula tell you about the subsets of a set of size \(n\text{?}\)
Hint.What is the sum on the left counting? What is the sum on the right counting?
