Factorial Using Recursion in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the factorial of a number?

The factorial of a number n (written as n!) is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24.

beginner

What is recursion in programming?

Recursion is when a function calls itself to solve smaller parts of a problem until it reaches a simple case it can solve directly.

beginner

What is the base case in a recursive factorial function?

The base case is when n equals 0 or 1, and the factorial is 1. This stops the recursion from continuing forever.

intermediate

How does the recursive factorial function calculate factorial(n)?

It multiplies n by factorial(n-1), calling itself with a smaller number until it reaches the base case.

intermediate

What happens if a recursive function does not have a base case?

The function will keep calling itself forever, causing a stack overflow error and crashing the program.

What is the factorial of 0?

A 1

B 0

C Undefined

D 0!

By definition, 0! is 1.

In a recursive factorial function, what does the function call factorial(n-1) represent?

A Returning the final answer

B Calling the function with a smaller problem

C Stopping the recursion

D Calling the function with the same problem

factorial(n-1) calls the function with a smaller number to break down the problem.

What is the base case for factorial recursion?

A n < 0

B n == 2

C n > 1

D n == 0 or n == 1

The base case is when n is 0 or 1, factorial is 1.

What will happen if the base case is missing in a recursive factorial function?

A Infinite recursion and crash

B Function returns 0

C Function returns 1

D Function runs normally

Without a base case, recursion never stops, causing a crash.

Which of these is the correct recursive formula for factorial?

A factorial(n) = factorial(n-1) - n

B factorial(n) = n + factorial(n-1)

C factorial(n) = n × factorial(n-1)

D factorial(n) = factorial(n-1) / n

Factorial multiplies n by factorial of (n-1).

Explain how a recursive function calculates the factorial of a number.

Think about how the function breaks the problem into smaller parts and stops.

Describe why a base case is important in recursion, especially in factorial calculation.

Without it, the function would never stop calling itself.

Practice

1. What does the factorial of a number represent in mathematics?

easy

A. The product of all positive integers up to that number

B. The sum of all positive integers up to that number

C. The number multiplied by 2

D. The number divided by 2

Solution

1. Step 1: Understand factorial definition

   Factorial of a number n (written as n!) is the product of all positive integers from 1 to n.

2. Step 2: Compare options with definition

   Only The product of all positive integers up to that number correctly states this product definition.

3. Final Answer:

   The product of all positive integers up to that number -> Option A

4. Quick Check:

   Factorial = product sequence [OK]

Hint: Factorial multiplies numbers down to 1 [OK]

Common Mistakes:

• Confusing factorial with sum
• Thinking factorial doubles the number
• Mixing factorial with division

2. Which of the following is the correct base case for a recursive factorial function in Python?

easy

A. if n == 1: return n + 1

B. if n == 1: return 0

C. if n == 0: return 0

D. if n == 0: return 1

Solution

1. Step 1: Identify base case for factorial

   Factorial of 0 is defined as 1, so base case should return 1 when n is 0.

2. Step 2: Check options for correct base case

   if n == 0: return 1 returns 1 when n == 0, which is correct. Others return wrong values.

3. Final Answer:

   if n == 0: return 1 -> Option D

4. Quick Check:

   Base case n=0 returns 1 [OK]

Hint: Base case factorial(0) = 1 [OK]

Common Mistakes:

• Returning 0 for factorial(0)
• Using wrong base case condition
• Returning n+1 instead of 1

3. What is the output of the following code?

def factorial(n):
    if n == 0:
        return 1
    else:
        return n * factorial(n-1)

print(factorial(4))

medium

A. 10

B. 16

C. 24

D. 120

Solution

1. Step 1: Trace recursive calls for factorial(4)

   factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 * factorial(0) factorial(0) = 1 (base case)

2. Step 2: Calculate values bottom-up

   factorial(1) = 1 * 1 = 1 factorial(2) = 2 * 1 = 2 factorial(3) = 3 * 2 = 6 factorial(4) = 4 * 6 = 24

3. Final Answer:

   24 -> Option C

4. Quick Check:

   4! = 24 [OK]

Hint: Multiply down from n to 1 recursively [OK]

Common Mistakes:

• Stopping recursion too early
• Adding instead of multiplying
• Wrong base case causing infinite recursion

4. Find the error in this recursive factorial function:

def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n+1)

print(factorial(3))

medium

A. The recursive call should use n-1, not n+1

B. The print statement should be outside the function

C. The function should return n + factorial(n-1)

D. The base case should be n == 0, not n == 1

Solution

1. Step 1: Analyze recursive call parameter

   The function calls factorial(n+1), which increases n each time, causing infinite recursion.

2. Step 2: Correct recursive call to decrement n

   It should call factorial(n-1) to move towards base case and stop recursion.

3. Final Answer:

   The recursive call should use n-1, not n+1 -> Option A

4. Quick Check:

   Recursion must approach base case by decreasing n [OK]

Hint: Recursive calls must reduce n to reach base case [OK]

Common Mistakes:

• Increasing n in recursive call causing infinite loop
• Wrong base case value
• Adding instead of multiplying in return

5. How can you modify the recursive factorial function to handle negative inputs gracefully?

hard

A. Change base case to n == -1

B. Add a check to return None or error if n < 0 before recursion

C. Use absolute value of n in recursive call

D. Ignore negative inputs and proceed as usual

Solution

1. Step 1: Understand factorial domain

   Factorial is not defined for negative numbers, so function should handle this case explicitly.

2. Step 2: Add input validation before recursion

   Check if n < 0 and return None or raise error to avoid invalid recursion.

3. Final Answer:

   Add a check to return None or error if n < 0 before recursion -> Option B

4. Quick Check:

   Validate input before recursion [OK]

Hint: Check for negative input before recursion [OK]

Common Mistakes:

• Using absolute value hides invalid input
• Changing base case to negative number
• Ignoring invalid inputs causes wrong results