Use factorial notation.

Simplify $\frac{8!}{6!}$.

Consider the expression $\frac{(n+2)!}{n!}$ where $n$ is a positive integer. (a) Use factorial notation to simplify this expression. (1 mark) (b) Hence determine the value of the expression when $n = 8$. (1 mark) (c) Use your result from part (a) to solve the equation $\frac{(k+2)!}{k!} = 132$ for positive integer $k$. (1 mark)

The table below shows values of a function $f(n)$ for selected positive integers $n$. Given that $f(n) = \frac{k \cdot (3n)!}{n! \cdot (2n)!}$ for some constant $k \in \mathbb{Z}^+$, determine the value of $k$, and hence evaluate $f(8)$.

Use factorial notation to express the product $7 \times 6 \times 5 \times 4$.

Determine the value of $n$ that satisfies the equation $\frac{(n+2)!}{n!} - \frac{(n+1)!}{(n-1)!} = 56$.