Factorial Using Recursion in DSA C# - Time & Space Complexity

Time Complexity: Factorial Using Recursion

O(n)

Understanding Time Complexity

We want to understand how the time to calculate a factorial grows as the number gets bigger.

How many steps does the computer take when using recursion to find factorial?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public int Factorial(int n)
{
    if (n <= 1) return 1;
    return n * Factorial(n - 1);
}
    

This code calculates the factorial of a number by calling itself with smaller numbers until it reaches 1.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive call to Factorial with n-1.
• How many times: Exactly n times until reaching the base case.

How Execution Grows With Input

Each time we call Factorial, it does one multiplication and calls itself with a smaller number.

Input Size (n)	Approx. Operations
10	About 10 multiplications and calls
100	About 100 multiplications and calls
1000	About 1000 multiplications and calls

Pattern observation: The number of steps grows directly with n, so doubling n doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to compute factorial grows in a straight line with the input number.

Common Mistake

[X] Wrong: "Recursion makes factorial calculation take exponential time because it calls itself many times."

[OK] Correct: Each recursive call reduces n by 1 and happens only once per number, so calls grow linearly, not exponentially.

Interview Connect

Understanding recursion time helps you explain how simple repeated steps add up, a key skill in many coding problems.

Self-Check

"What if we used a loop instead of recursion? How would the time complexity change?"

Practice

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

easy

A. The number multiplied by 2

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

C. The number divided by 2

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

Solution

1. Step 1: Understand factorial definition

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

2. Step 2: Compare options

   Only The product of all positive integers up to that number correctly describes factorial as a product, not sum or division.

3. Final Answer:

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

4. Quick Check:

   Factorial = product of integers [OK]

Hint: Factorial multiplies numbers down to 1 [OK]

Common Mistakes:

• Confusing factorial with sum
• Thinking factorial divides the number
• Assuming factorial doubles the number

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

easy

A. if (n == 0) return 1;

B. if (n == 1) return 1;

C. if (n == 2) return 2;

D. if (n == -1) return 1;

Solution

1. Step 1: Identify factorial base case

   By definition, 0! = 1, so the base case should return 1 when n is 0.

2. Step 2: Check options

   Only if (n == 0) return 1; returns 1 when n == 0, which is correct base case for recursion.

3. Final Answer:

   if (n == 0) return 1; -> Option A

4. Quick Check:

   Base case: 0! = 1 [OK]

Hint: Base case for factorial is n == 0 returning 1 [OK]

Common Mistakes:

• Using n == 1 as base case
• Returning wrong value in base case
• Not handling negative inputs

3. What is the output of the following C# code?

static int Factorial(int n) {
    if (n == 0) return 1;
    return n * Factorial(n - 1);
}

Console.WriteLine(Factorial(4));

medium

A. 24

B. 10

C. 16

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 A

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
• Misunderstanding base case

4. Identify the error in this recursive factorial function:

static int Factorial(int n) {
    if (n == 1) return 1;
    return n * Factorial(n);
}

medium

A. Return type should be void

B. Missing base case for n == 0

C. Recursive call uses same n, causing infinite recursion

D. Multiplication operator is incorrect

Solution

1. Step 1: Analyze recursive call

   The function calls Factorial(n) inside itself without changing n, causing infinite recursion.

2. Step 2: Identify fix

   Recursive call should be Factorial(n - 1) to reduce n and reach base case.

3. Final Answer:

   Recursive call uses same n, causing infinite recursion -> Option C

4. Quick Check:

   Recursive call must reduce n [OK]

Hint: Recursive call must change parameter to approach base case [OK]

Common Mistakes:

• Calling function with same argument
• Forgetting base case
• Wrong return type

5. How would you modify the recursive factorial function to handle negative inputs by returning -1 for any negative number?

static int Factorial(int n) {
    if (n == 0) return 1;
    return n * Factorial(n - 1);
}

hard

A. No change needed; factorial works for negatives

B. Add if (n < 0) return -1; before base case

C. Add if (n == 1) return -1; after base case

D. Change base case to if (n <= 0) return 1;

Solution

1. Step 1: Understand negative input handling

   Factorial is undefined for negative numbers, so function should return -1 for n < 0.

2. Step 2: Modify function

   Add a check at start: if (n < 0) return -1; before base case to handle negatives properly.

3. Final Answer:

   Add if (n < 0) return -1; before base case -> Option B

4. Quick Check:

   Return -1 for negatives before recursion [OK]

Hint: Check negatives first, return -1 before recursion [OK]

Common Mistakes:

• Changing base case incorrectly
• Not handling negatives at all
• Returning wrong value for negatives