2. Let $k$ and $n$ be positive integers such that $n \geq 2 k$ and $$\frac { ( n - 2 ) ! } { ( n - 2 k ) ! } = k ! 2 ^ { k - 1 }$$ (Recall that, for $r \geq 1 , r !$ is the product $r \cdot ( r - 1 ) \cdot ( r - 2 ) \ldots 2.1$ and that $0 !$ is defined to equal 1.) (i) Suppose that $k = 1$. What are the possible values of $n$ ? (ii) Suppose that $k = 2$. Show that $( n - 2 ) ( n - 3 ) = 4$. What are the possible values of $n$ ? (iii) Suppose that $k = 3$. Show that it is impossible that $n \geq 7$. (iv) Suppose that $k \geq 4$. Show that there are no possible values of $n$.
2. Let $k$ and $n$ be positive integers such that $n \geq 2 k$ and
$$\frac { ( n - 2 ) ! } { ( n - 2 k ) ! } = k ! 2 ^ { k - 1 }$$
(Recall that, for $r \geq 1 , r !$ is the product $r \cdot ( r - 1 ) \cdot ( r - 2 ) \ldots 2.1$ and that $0 !$ is defined to equal 1.)\\
(i) Suppose that $k = 1$. What are the possible values of $n$ ?\\
(ii) Suppose that $k = 2$. Show that $( n - 2 ) ( n - 3 ) = 4$. What are the possible values of $n$ ?\\
(iii) Suppose that $k = 3$. Show that it is impossible that $n \geq 7$.\\
(iv) Suppose that $k \geq 4$. Show that there are no possible values of $n$.\\