Chapter 11.4, Problem 46ES

To determine

(a)

To prove:

Show that $n!\text{ is }O\left(n^n\right).$

To determine

(b)

Use part (a) to show that ${\log }_2\left(n!\right)\text{ is }O\left(n{\log }_{2}n\right).$

To determine

(c)

Show that $n^n\le {\left(n!\right)}^2\text{ for all positive integers }n\ge 2.$

To determine

(d)

Use part (c) to show that ${\log }_2\left(n!\right)\text{ is }\Omega \left(n{\log }_{2}n\right).$

To determine

(e)

Use part (b) and (d) to find an order for ${\log }_2\left(n!\right).$