What do you think $0!$ is equal to?  

1. Try it on your calculator to see what you get.

   Answer (a):

   $1$

2. What does $_8P_8$ mean? What should $_8P_8$ be equal to? Write $_8P_8$ using the factorial formula. Why is it necessary for $0!$ to equal $1$?

   Hint (b):

   Refer to the Math Notes box below for help.

3. Do you remember how to show that $2^0 = 1$? You can use a sequence of powers of two like this: $\frac { 2 ^ { 4 } } { 2 } = 2^3$, $\frac { 2 ^ { 3 } } { 2 }= 2^2$, $\frac { 2 ^ { 2 } } { 2 }= 2^1$, so $\frac { 2 ^ { 1 } } { 2 } = 2^0$. Since $\frac { 2 ^ { 1 } } { 2 }= 1$ you also know that $2^0 = 1$.
   You can construct a similar pattern for $0!$, starting with $\frac { 5 ! } { 5 }= 4!$ and then $\frac { 4 ! } { 4 }= 3!$ . Continue the pattern and make an argument to justify that $0! = 1$.

   Answer (c):

   $\frac{3!}{3} - 2!$, $\frac{2!}{2} = 1$, $\frac{1!}{1} = 0!$

   $\frac{1!}{1} = \frac{1}{1} = 1 = 0!$