A student wants to know$$ why, if we can define$0!$ as$1,$ we cannot define

StartFraction $1$ Over $0$ EndFraction$10$

as $1.$ How do you$$ respond?

Question content area bottom

Part $1$

Choose the correct answer below.

A$.$

$0!$ is defined as $1$ because there are no ways to arrange zero objects. Defining

StartFraction $1$ Over $0$ EndFraction$10$

as $1$ would cause large inconsistencies.

B$.$

$0!$ never comes up so nothing changes if we define it to be$1,$ but

StartFraction $1$ Over $0$ EndFraction$10$

happens a lot so it cannot be defined for convenience.

C$.$

Defining

StartFraction $1$ Over $0$ EndFraction$10$

as $1$ can be done in some situations.$$

D$.$

$0!$ is defined as $1$ because it fits the formula for combinations and is consistent with the rest of mathematics. Defining

StartFraction $1$ Over $0$ EndFraction$10$

as $1$ would cause large inconsistencies.