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Chapter 22 Binomial Theorem
Exercises Practice Problems
1. Nested Sets.
In this problem, you will prove the following identity in two different ways
\begin{equation*}
3^n = \sum_{k=0}^n 2^k { n \choose k}
\end{equation*}
Prove this equation using the binomial theorem.
-
Give a combinatorial proof of this equation by showing that each side counts the number of ways to choose sets \(A,B\) where \(\emptyset \subseteq A \subseteq B \subseteq [n]\text{.}\)
The left hand side is \(3^n\text{.}\) Explain why there are three choices for where to place every element of \([n]\text{.}\)
The right hand is \(\sum_{k=0}^n 2^k { n \choose k}.\) This time, we create the sets \(A \subseteq B\) in a different way. Namely, first we decide what the size of \(B\) will be, and use \(k\) to denote this size. With this in mind, interpret the right hand side of this equation.
2. Rectangles Galore.
The following \(2 \times 2\) grid contains 9 rectangles (many of them are squares). Draw them all.
Prove that an \(m \times n\) grid contains \({m+1 \choose 2}{n+1 \choose 2}\) rectangles.
3. Diagonals in an \(n\)-gon.
Consider a convex \(n\)-gon with its vertices labeled by the elements of \([n]\text{.}\)
How many distinct diagonals are are there?
How may pairs of crossing diagonals are there?
