grade lớp 11 level chuyên để difficulty easy updated 04/10/2026

Factorial

Abstract

This lesson presents the factorial calculation problem using recursion.

Problem statement

Task:
Using recursion, write a program to calculate \(n!\).

Given that \(0! = 1\) and \(1! = 1\).

Input:
A non-negative integer \(n\).

Output:
An integer representing the factorial of \(n\).

Test cases:

No.	Input	Output
1	0	1
2	1	1
3	6	720
4	21	51090942171709440000

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Proposed solution

Core idea

We have the recurrence relation for calculating the factorial as follows:

\[n! = \begin{cases} 1 & \text{if } n = 0 \text{ or } n = 1 \\ (n - 1)! \times n & \text{if } n > 1 \\ \end{cases}\]

According to the formula above:

1. Base case: \(n = 0\) or \(n = 1\).

   No further recursive calls are made; the function simply returns 1.

2. Recursive case: all remaining values of \(n\) (assuming \(n\) is not a negative integer).

   The function calls itself with n - 1 as the parameter to calculate \((n - 1)!\), and then multiplies the result by \(n\).

   Example:
   To calculate \(5!\), the function calls itself to compute \((5 - 1)! = 4!\), and then multiplies the result by \(5\).

   Specifically:

   • The factorial(5) function recursively calls factorial(4).
   • The factorial(4) function recursively calls factorial(3).
   • The factorial(3) function recursively calls factorial(2).
   • The factorial(2) function recursively calls factorial(1).
   • The factorial(1) function makes no further recursive calls and returns 1.

Writing the program

1. Write the factorial() function with a single parameter n.

def factorial(n):
    # Trường hợp cơ sở
    if n == 0 or n == 1:
        return 1

    # Trường hợp đệ quy
    return factorial(n - 1) * n

2. Write the main program:

• Prompt the user to enter a non-negative integer and store it in the variable number.
• Call the factorial() function with number as the argument and store the returned value in the variable result.
• Print the result.

if __name__ == '__main__':
    number = int(input('Nhập số nguyên không âm: '))

    result = factorial(number)
    print(f'{number}! = {result}')

3. Run the program and enter 6; the output is as follows:

Nhập n nguyên dương: 6
6! = 720

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Source code

The complete source code is available at:

• Google Colab
• GitHub