Python Tutorial

Introduction Python Features Python Applications System requirements for Python Python Installation Python Data Types Python IDE Python Keywords Python Operators Python Comments Python Pass Statement

Python Conditional Statements

Python if Statement Python elif Statement Python Switch Case

Python Loops

Python for loop Python while loop Python Break Statement Python Continue Statement Python Goto Statement

Python Arrays

Python Array Python Matrix

Python Strings

Python Strings Python Regex

Python Built-in Data Structure

Python Lists Python Tuples Python Dictionary Python Sets

Python Functions

Python Function Python min() function Python max() function Python User-define Functions Python Built-in Functions Anonymous/Lambda Function in Python

Python File Handling

Python File Handling Python Read CSV Python Write CSV Python Read Excel Python Write Excel Python Read Text File Python Write Text File Read JSON File in Python

Python Exception Handling

Python Exception Handling Python Errors and exceptions Python Assert

Python OOPs Concept

OOPs Concepts in Python Inheritance in Python Polymorphism in Python Python Encapsulation Python Constructor

Python Iterators

Iterators in Python Yield Statement In Python

Python Generators

Python Generator

Python Decorators

Python Decorator

Python Functions and Methods

Python Built-in Functions Python String Methods Python List Methods Python Dictionary Methods Python Tuple Methods Python Set Methods

Python Modules

Python Modules Python Datetime Module Python Calendar Module  

Python MySQL

Python MySQL Python MySQL Update Operation Python MySQL Delete Operation

Python MongoDB

Python MongoDB

Python Data Structure Implementation

Python Stack Python Queue Python Hash Table Python Graph

Python Advance Topics

Speech Recognition in Python Face Recognition in Python Python Rest API Python Command Line Arguments Python JSON Python Virtual Environment Type Casting in Python Collections in python Python Enumerate Python Debugger Python DefaultDict


Python PPTX Python Pickle Python Seaborn Python Coroutine Python EOL Python Infinity Python math.cos and math.acos function Python Project Ideas Based On Django Reverse a String in Python Reverse a Number in Python Python Word Tokenizer Python Trigonometric Functions Python try catch exception GUI Calculator in Python Implementing geometric shapes into the game in python Installing Packages in Python Python Try Except Python Sending Email Socket Programming in Python Python CGI Programming Python Data Structures Python abstract class Python Compiler Python K-Means Clustering List Comprehension in Python3 NSE Tools In Python Operator Module In Python Palindrome In Python Permutations in Python Pillow Python introduction and setup Python Functionalities of Pillow Module Python Argmin Python whois Python JSON Schema Python lock Return Statement In Python Reverse a sentence In Python tell() function in Python Why learn Python? Write Dictionary to CSV in Python Write a String in Python Binary Search Visualization using Pygame in Python Latest Project Ideas using Python 2022 Closest Pair of Points in Python ComboBox in Python Python vs R Python Ternary Operators Self in Python Python vs Java Python Modulo Python Packages Python Syntax Python Uses Python Logical Operators Python Multiprocessing Python History Difference between Input() and raw_input() functions in Python Conditional Statements in python Confusion Matrix Visualization Python Python Algorithms Python Modules List Difference between Python 2 and Python 3 Is Python Case Sensitive Method Overloading in Python Python Arithmetic Operators Design patterns in python Assignment Operators in Python Is Python Object Oriented Programming language Division in Python Python exit commands Continue And Pass Statements In Python Colors In Python Convert String Into Int In Python Convert String To Binary In Python Convert Uppercase To Lowercase In Python Convert XML To JSON In Python Converting Set To List In Python Covariance In Python CSV Module In Python Decision Tree In Python Difference Between Yield And Return In Python Dynamic Typing In Python Abstract design pattern in python Builder design pattern in python

How to

How to convert integer to float in Python How to reverse a string in Python How to take input in Python How to install Python in Windows How to install Python in Ubuntu How to install PIP in Python How to call a function in Python How to download Python How to comment multiple lines in Python How to create a file in Python How to create a list in Python How to declare array in Python How to clear screen in Python How to convert string to list in Python How to take multiple inputs in Python How to write a program in Python How to compare two strings in Python How to create a dictionary in Python How to create an array in Python How to update Python How to compare two lists in Python How to concatenate two strings in Python How to print pattern in Python How to check data type in python How to slice a list in python How to implement classifiers in Python How To Print Colored Text in Python How to develop a game in python How to print in same line in python How to create a class in python How to find square root in python How to import numy in python How to import pandas in python How to uninstall python How to upgrade PIP in python How to append a string in python How to open a file in python


Python Sort List Sort Dictionary in Python Python sort() function Python Bubble Sort


Factorial Program in Python Prime Number Program in Python Fibonacci Series Program in Python Leap Year Program in Python Palindrome Program in Python Check Palindrome In Python Calculator Program in Python Armstrong Number Program in Python Python Program to add two numbers Anagram Program in Python Even Odd Program in Python GCD Program in Python Python Exit Program Python Program to check Leap Year Operator Overloading in Python Pointers in Python Python Not Equal Operator Raise Exception in Python Salary of Python Developers in India What is a Script in Python Singleton design pattern in python

Python Pascal Triangle

Python Pascal Triangle

Pascal triangle

A pascal triangle is a number pattern of triangular array of the binomial coefficients. For designing a pascal triangle, we write a function in the program which takes an integer value as the input and print the number of lines (given by user as the integer value) for the pascal triangle.

Example: Following is the example of a pascal triangle pattern with the first 6 rows:

 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1
 1 5 10 10 5 1 

Pascal triangle in Python

In Python, we draw the pascal triangle pattern using the math module. We use the factorial functions of the math module in our Python program to implement the nCr formula for pascal triangle.

Here, in this tutorial, we will learn about the following methods in our Python program to get the pascal triangle pattern in the output:

1). Directly implementing nCr formula

2). Using C (line, m-1) implementation

3). Using powers of 11 implementation

Let's learn about the all the above given methods details and understand them with an example program.

Method 1: Directly implementing the nCr Formula in the program:

When we use the nCr formula in our program, a pictorial representation as given below will appear on the screen:


        ¹C0   ¹C1



We will use the following steps or algorithm in the program while implementing this method:

  • We will first take an input integer which is the number of rows of pascal triangle to be printed. Let's assume this input integer as 'num'.
  • We will make outer iteration of m variable from 0 to given num times, for printing the rows of triangle.
  • Then, we will make inner iteration of variable n from 0 to (num-1) times, for printing variables in each row.
  • After that, we will print single blank space as " ".
  • Then, we will close the inner loop (n variable loop) and it will create the left spacing in the rows.
  • After that, we will make an inner iteration for the n variable from 0 to m.
  • Then, we will print the nCr result for m and n.
  • Now, we will close the inner loop of program.
  • In last, we will print or add new line after each inner iteration of n loop. It will add a new line while printing a new row.

Now, look at the following program for the implementation of above algorithm:

Example –

 # importing factorial functions from math module
 from math import factorial
 # taking number of rows as input from user
 num = int (input ("Enter the number of rows for pascal triangle: "))
 # defining the outer iteration for number of loops
 for m in range(num):
         # defining inner iteration for variables in each row
             for n in range(num-m+1):
                         # closing inner n iteration for left spacing
                         print (end=" ")
         # inner n iteration for elements of pascal triangle
             for n in range(m+1):
                         # implementing the nCr = n!/((n-r)!*r!) formula
                         print (factorial(m)//(factorial(n)*factorial(m-n)), end=" ")
             # using print statement for new line
             print () 


 Enter the number of rows for pascal triangle: 6
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1
   1 5 10 10 5 1
 The time complexity for the pascal triangle we have printed above is O(N²). 

Method 2: Using C (line, m-1) implementation:

In this method, we will learn that how we can optimize the complexity of code given in method 1. In this method, we will follow the concept of binomial coefficients i.e., the mth entry in a given line number (let's say line) is the binomial coefficient for C (line, m). And, all the lines of pascal triangle will start with the value 1.

The basic idea used in this implementation is that we have to calculate the C (line, m) binomial coefficients using the C (line, m-1) coefficient. General formula for such type of implementation is as follows:

C (line, m) = [C (line, m-1) * (line - m + 1)]/ m

Now, look at the following program for the implementation of above given method:

Example – 

 # taking number of rows as input from user
 num = int (input ("Enter the number of rows for pascal triangle: "))
 # defining for loop for number of rows
 for m in range(1, num+1):
 # using inner iteration for elements in triangle rows
 for n in range(0, num-m+1):
         # for left spacing in rows
 print (' ', end='')
 # defining C as first element of array equals to 1
     C = 1
     # using for loop for values in line
 for n in range(1, m+1):
         # the first value in a line is always equals to 1
 print (' ', C, sep='', end='')
      # using Binomial Coefficient for C variable
         C = C * (m - n) // n
 # printing new line for each row


Enter the number of rows for pascal triangle: 7         
        1 1
       1 2 1
      1 3 3 1
     1 4 6 4 1
    1 5 10 10 5 1
   1 6 15 20 15 6 1
 The time complexity for the pascal triangle we have printed above is O(N²). 

Method 3: Using powers of 11 implementation:

This method is considered as the most optimized approach for printing pascal triangle pattern in Python. This method is based on the approach of powers of 11. Look at the following powers of 11 to understand this approach:

 11? = 1
 11¹ = 11
 11² = 121
 11³ = 1331 etc. 

Note: However, this approach is only limited up to the n = 5 i.e., 11?. It means we cannot print more than 5 rows of pascal triangle using this approach.

Now, look at the following program for the implementation of above given approach:

Example – 1

 # taking number of rows as input from user
 num = int (input ("Enter the number of rows for pascal triangle (Maximum 5): "))
 # using for loop for number of rows
 for a in range(num):
     # adjust left spacing between each element
 print (' '*(num-a), end='')
     # computing power of 11 for each row
 print(' '.join(map(str, str(11**a)))) 


 Enter the number of rows for pascal triangle (Maximum 5): 5
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1
 The time complexity for the pascal triangle we have printed above is O(N).