5) What is the base case for the recursive definition of factorial?a) 0! = 1b) 1! = 1c) 0! = 0d) 1! = 0

Break Down the Problem

1. Identify the definition of factorial.
2. Determine what a base case in a recursive definition implies.
3. Evaluate each of the given options in relation to the definition of factorial.

Relevant Concepts

• The factorial of a non-negative integer $n$ is defined as:
  • $n! = n \times (n-1)!$ for $n > 0$
  • The base case is crucial for stopping the recursion and defining the function for 0.

Analysis and Detail

1. Factorial Definition:

   • $n! = n \times (n-1)!$
   • The recursion continues until $n = 0$.

2. Base Cases:

   • For factorials, the base case is defined as $0! = 1$ which is a classic definition in mathematics.

3. Evaluating Options:

   • a) $0! = 1$: This is indeed the correct base case.
   • b) $1! = 1$: While correct, it doesn't serve as the base case for the recursive factorial definition.
   • c) $0! = 0$: Incorrect, as $0!$ equals 1.
   • d) $1! = 0$: Incorrect as $1! = 1$.

Verify and Summarize

The correct base case for the recursion defining factorial corresponds to the value of $0!$ being equal to 1. This allows the recursive function to stop at the non-negative integer zero.

Final Answer

The base case for the recursive definition of factorial is: a) $0! = 1$.