91Ó°ÊÓ

What is the coefficient of \(x^{5} y^{13}\) in the expansion of \((3 x-2 y)^{18} ?\) What is the coefficient of \(x^{8} y^{9} ?\) (There is not a misprint in this last question!)

By integrating the binomial expansion, prove that, for a positive integer \(n\), $$ 1+\frac{1}{2}\left(\begin{array}{l} n \\ 1 \end{array}\right)+\frac{1}{3}\left(\begin{array}{l} n \\ 2 \end{array}\right)+\cdots+\frac{1}{n+1}\left(\begin{array}{l} n \\ n \end{array}\right)=\frac{2^{n+1}-1}{n+1} $$.

Prove that, for all real numbers \(r\) and all integers \(k\) and \(m\), $$ \left(\begin{array}{c} r \\ m \end{array}\right)\left(\begin{array}{l} m \\ k \end{array}\right)=\left(\begin{array}{l} r \\ k \end{array}\right)\left(\begin{array}{c} r-k \\ m-k \end{array}\right) . $$

Every day a student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She always takes a shortest walk of 24 blocks. (a) How many different walks are possible? (b) Suppose that four blocks east and five blocks north of her home lives her best friend, whom she meets each day on her way to school. Now how many different walks are possible? (c) Suppose, in addition, that three blocks east and six blocks north of her friend's house there is a park where the two girls stop each day to rest and play. Now how many different walks are there? (d) Stopping at a park to rest and play, the two students often get to school late. To avoid the temptation of the park, our two students decide never to pass the intersection where the park is. Now how many different walks are there?

Let \(n\) and \(k\) be positive integers. Give a combinatorial proof of the identity \((5.15)\) $$ n(n+1) 2^{n-2}=\sum_{k=1}^{n} k^{2}\left(\begin{array}{l} n \\ k \end{array}\right) . $$

Use the multinomial theorem to show that, for positive integers \(n\) and \(t\). $$ t^{n}=\sum\left(\begin{array}{c} n \\ n_{1} n_{2} \cdots n_{t} \end{array}\right) $$, where the summation extends over all nonnegative integral solutions \(n_{1}, n_{2}, \ldots, n_{t}\) of \(n_{1}+n_{2}+\cdots+n_{t}=n\).

Use the multinomial theorem to expand \(\left(x_{1}+x_{2}+x_{3}\right)^{4}\).

Prove that $$ \sum_{n_{1}+n_{2}+n_{3}=n}\left(\begin{array}{cc} n & \\ n_{1} n_{2} n_{3} \end{array}\right)(-1)^{n_{1}-n_{2}+n_{3}}=1 $$ where the summation extends over all nonnegative integral solutions of \(n_{1}+n_{2}+\) \(n_{3}=n .\)

Consider the partially ordered set \((X, \mid)\) on the set \(X=\\{1,2, \ldots, 12\\}\) of the first 12 positive integers, partially ordered by "is divisible by." (a) Determine a chain of largest size and a partition of \(X\) into the smallest. number of antichains. (b) Determine an antichain of largest size and a partition of \(X\) into the smallest number of chains.