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Pascal’s Triangle

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Pascal’s triangle is a triangular array of binomial coefficients. Write a function that takes an integer value N as input and prints the first N lines of Pascal’s triangle.

Example:

The below image shows the Pascal’s Triangle for N=6

Pascal's-Triangle-for-first-6-Rows

Recommended Practice

Pascal’s Triangle using Binomial Coefficient:

The number of entries in every line is equal to line number. For example, the first line has “1, the second line has “1 1“, the third line has “1 2 1“,.. and so on. Every entry in a line is value of a Binomial Coefficient. The value of ith entry in line number line is C(line, i). The value can be calculated using following formula. 

  • C(line, i) = line! / ( (line-i)! * i! )
 

Algorithm:

  • Run a loop for each row of pascal’s triangle i.e. 1 to N.
    • For each row, run an internal loop for each element of that row.
      • Calculate the binomial coefficient for the element using the formula mentioned in the approach.

Below is the implementation of the above approach:

C++




//  C++ code for Pascal's Triangle
#include <iostream>
using namespace std;
 
 
int binomialCoeff(int n, int k);
 
// Function to print first
// n lines of Pascal's
// Triangle
void printPascal(int n)
{
    // Iterate through every line and
    // print entries in it
    for (int line = 0; line < n; line++) {
        // Every line has number of
        // integers equal to line
        // number
        for (int i = 0; i <= line; i++)
            cout << " " << binomialCoeff(line, i);
        cout << "\n";
    }
}
 
// See
// for details of this function
int binomialCoeff(int n, int k)
{
    int res = 1;
    if (k > n - k)
        k = n - k;
    for (int i = 0; i < k; ++i) {
        res *= (n - i);
        res /= (i + 1);
    }
 
    return res;
}
 
// Driver program
int main()
{
    int n = 7;
    printPascal(n);
    return 0;
}
 
// This code is contributed by shivanisinghss2110


C




//  C++ code for Pascal's Triangle
#include <stdio.h>
 
// for details of this function
int binomialCoeff(int n, int k);
 
// Function to print first
// n lines of Pascal's
// Triangle
void printPascal(int n)
{
    // Iterate through every line and
    // print entries in it
    for (int line = 0; line < n; line++)
    {
        // Every line has number of
        // integers equal to line
        // number
        for (int i = 0; i <= line; i++)
            printf("%d ",
                    binomialCoeff(line, i));
        printf("\n");
    }
}
 
// for details of this function
int binomialCoeff(int n, int k)
{
    int res = 1;
    if (k > n - k)
    k = n - k;
    for (int i = 0; i < k; ++i)
    {
        res *= (n - i);
        res /= (i + 1);
    }
     
    return res;
}
 
// Driver program
int main()
{
    int n = 7;
    printPascal(n);
    return 0;
}


Java




// Java code for Pascal's Triangle
import java.io.*;
 
class GFG {
     
    // Function to print first
    // n lines of Pascal's Triangle
    static void printPascal(int n)
    {
         
    // Iterate through every line
    // and print entries in it
    for (int line = 0; line < n; line++)
    {
        // Every line has number of
        // integers equal to line number
        for (int i = 0; i <= line; i++)
        System.out.print(binomialCoeff
                        (line, i)+" ");
                         
        System.out.println();
    }
    }
     
    // Link for details of this function
    static int binomialCoeff(int n, int k)
    {
        int res = 1;
         
        if (k > n - k)
        k = n - k;
             
        for (int i = 0; i < k; ++i)
        {
            res *= (n - i);
            res /= (i + 1);
        }
        return res;
    }
     
    // Driver code
    public static void main(String args[])
    {
    int n = 7;
    printPascal(n);
    }
}
 
/*This code is contributed by Nikita Tiwari.*/


Python3




# Python 3 code for Pascal's Triangle
# A simple O(n^3)
# program for
# Pascal's Triangle
 
# Function to print
# first n lines of
# Pascal's Triangle
def printPascal(n) :
     
    # Iterate through every line
    # and print entries in it
    for line in range(0, n) :
         
        # Every line has number of
        # integers equal to line
        # number
        for i in range(0, line + 1) :
            print(binomialCoeff(line, i),
                " ", end = "")
        print()
     
 
# for details of this function
def binomialCoeff(n, k) :
    res = 1
    if (k > n - k) :
        k = n - k
    for i in range(0 , k) :
        res = res * (n - i)
        res = res // (i + 1)
     
    return res
 
# Driver program
n = 7
printPascal(n)
 
 
# This code is contributed by Nikita Tiwari.


C#




// C# code for Pascal's Triangle
using System;
 
class GFG {
     
    // Function to print first
    // n lines of Pascal's Triangle
    static void printPascal(int n)
    {
         
    // Iterate through every line
    // and print entries in it
    for (int line = 0; line < n; line++)
    {
        // Every line has number of
        // integers equal to line number
        for (int i = 0; i <= line; i++)
        Console.Write(binomialCoeff
                        (line, i)+" ");
                         
        Console.WriteLine();
    }
    }
     
    // Link for details of this function
    static int binomialCoeff(int n, int k)
    {
        int res = 1;
         
        if (k > n - k)
        k = n - k;
             
        for (int i = 0; i < k; ++i)
        {
            res *= (n - i);
            res /= (i + 1);
        }
        return res;
    }
     
    // Driver code
    public static void Main()
    {
    int n = 7;
    printPascal(n);
    }
}
 
/*This code is contributed by vt_m.*/


Javascript




<script>
 
// Javascript code for Pascal's Triangle
 
    // Function to print first
    // n lines of Pascal's Triangle
    function printPascal(n)
    {
         
    // Iterate through every line
    // and print entries in it
    for (let line = 0; line < n; line++)
    {
        // Every line has number of
        // integers equal to line number
        for (let i = 0; i <= line; i++)
        document.write(binomialCoeff
                        (line, i)+" ");
                         
        document.write("<br />");
    }
    }
     
    // Link for details of this function
    function binomialCoeff(n, k)
    {
        let res = 1;
         
        if (k > n - k)
        k = n - k;
             
        for (let i = 0; i < k; ++i)
        {
            res *= (n - i);
            res /= (i + 1);
        }
        return res;
    }
 
 
// Driver Code
 
    let n = 7;
    printPascal(n);
 
</script>


PHP




<?php
// PHP implementation for
// Pascal's Triangle
 
// for details of this function
function binomialCoeff($n, $k)
{
    $res = 1;
    if ($k > $n - $k)
    $k = $n - $k;
    for ($i = 0; $i < $k; ++$i)
    {
        $res *= ($n - $i);
        $res /= ($i + 1);
    }
return $res;
}
 
// Function to print first
// n lines of Pascal's
// Triangle
function printPascal($n)
{
    // Iterate through every line and
    // print entries in it
    for ($line = 0; $line < $n; $line++)
    {
        // Every line has number of
        // integers equal to line
        // number
        for ($i = 0; $i <= $line; $i++)
                echo "".binomialCoeff($line, $i)." ";
                 
        echo "\n";
    }
}
 
// Driver Code
$n=7;
printPascal($n);
 
// This code is contributed by Mithun Kumar
?>


Output

 1
 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

Time complexity: O(N^3), where N is the number of rows you want to print
Auxiliary Space: O(1)

Pascal’s Triangle using Dynamic Programming:

If we take a closer at the triangle, we observe that every entry is sum of the two values above it. So using dynamic programming we can create a 2D array that stores previously generated values. In order to generate a value in a line, we can use the previously stored values from array. 

Cases:

  • If line == 0 or line == i
    • arr[line][i] =1
  • Else:
    • arr[line][i] = arr[line-1][i-1] + arr[line-1][i]

Below is the implementation of the above approach:

C++




// C++ program for Pascal’s Triangle
// A O(n^2) time and O(n^2) extra space
// method for Pascal's Triangle
#include <bits/stdc++.h>
using namespace std;
 
void printPascal(int n)
{
     
    // An auxiliary array to store
    // generated pascal triangle values
    int arr[n][n];
 
    // Iterate through every line and
    // print integer(s) in it
    for (int line = 0; line < n; line++)
    {
        // Every line has number of integers
        // equal to line number
        for (int i = 0; i <= line; i++)
        {
        // First and last values in every row are 1
        if (line == i || i == 0)
            arr[line][i] = 1;
           
        // Other values are sum of values just
        // above and left of above
        else
            arr[line][i] = arr[line - 1][i - 1] +
                            arr[line - 1][i];
        cout << arr[line][i] << " ";
        }
        cout << "\n";
    }
}
 
// Driver code
int main()
{
    int n = 5;
    printPascal(n);
    return 0;
}
 
// This code is Contributed by Code_Mech.


C




// C program for Pascal’s Triangle
// A O(n^2) time and O(n^2) extra space
// method for Pascal's Triangle
void printPascal(int n)
{
// An auxiliary array to store
// generated pascal triangle values
int arr[n][n];
 
// Iterate through every line and print integer(s) in it
for (int line = 0; line < n; line++)
{
    // Every line has number of integers
    // equal to line number
    for (int i = 0; i <= line; i++)
    {
    // First and last values in every row are 1
    if (line == i || i == 0)
        arr[line][i] = 1;
    // Other values are sum of values just
    // above and left of above
    else
        arr[line][i] = arr[line-1][i-1] + arr[line-1][i];
    printf("%d ", arr[line][i]);
    }
    printf("\n");
}
}
// Driver code
int main()
{
int n = 5;
    printPascal(n);
    return 0;
}


Java




// java program for Pascal's Triangle
// A O(n^2) time and O(n^2) extra
// space method for Pascal's Triangle
import java.io.*;
 
class GFG {
    public static void main (String[] args) {
        int n = 5;
        printPascal(n);
    }
 
public static void printPascal(int n)
{
// An auxiliary array to store generated pascal triangle values
int[][] arr = new int[n][n];
 
// Iterate through every line and print integer(s) in it
for (int line = 0; line < n; line++)
{
    // Every line has number of integers equal to line number
    for (int i = 0; i <= line; i++)
    {
    // First and last values in every row are 1
    if (line == i || i == 0)
        arr[line][i] = 1;
    else // Other values are sum of values just above and left of above
        arr[line][i] = arr[line-1][i-1] + arr[line-1][i];
    System.out.print(arr[line][i]);
    }
    System.out.println("");
}
}
}


Python3




# Python3 program for Pascal's Triangle
 
# A O(n^2) time and O(n^2) extra
# space method for Pascal's Triangle
def printPascal(n:int):
 
    # An auxiliary array to store
    # generated pascal triangle values
    arr = [[0 for x in range(n)]
              for y in range(n)]
 
    # Iterate through every line
    # and print integer(s) in it
    for line in range (0, n):
 
        # Every line has number of
        # integers equal to line number
        for i in range (0, line + 1):
 
            # First and last values
            # in every row are 1
            if(i is 0 or i is line):
                arr[line][i] = 1
                print(arr[line][i], end = " ")
 
            # Other values are sum of values
            # just above and left of above
            else:
                arr[line][i] = (arr[line - 1][i - 1] +
                                arr[line - 1][i])
                print(arr[line][i], end = " ")            
        print("\n", end = "")
 
# Driver Code
n = 5
printPascal(n)
 
# This code is contributed
# by Sanju Maderna


C#




// C# program for Pascal's Triangle
// A O(n^2) time and O(n^2) extra
// space method for Pascal's Triangle
using System;
 
class GFG
{
public static void printPascal(int n)
{
     
// An auxiliary array to store
// generated pascal triangle values
int[,] arr = new int[n, n];
 
// Iterate through every line
// and print integer(s) in it
for (int line = 0; line < n; line++)
{
    // Every line has number of
    // integers equal to line number
    for (int i = 0; i <= line; i++)
    {
         
    // First and last values
    // in every row are 1
    if (line == i || i == 0)
        arr[line, i] = 1;
    else // Other values are sum of values
         // just above and left of above
        arr[line, i] = arr[line - 1, i - 1] +
                       arr[line - 1, i];
    Console.Write(arr[line, i]);
    }
Console.WriteLine("");
}
}
 
// Driver Code
public static void Main ()
{
    int n = 5;
    printPascal(n);
}
}
 
// This code is contributed
// by Akanksha Rai(Abby_akku)


Javascript




<script>
 
// javascript program for Pascal's Triangle
// A O(n^2) time and O(n^2) extra
// space method for Pascal's Triangle
var n = 5;
printPascal(n);
 
function printPascal(n)
{
// An auxiliary array to store generated pascal triangle values
arr = a = Array(n).fill(0).map(x => Array(n).fill(0));
 
// Iterate through every line and print integer(s) in it
for (line = 0; line < n; line++)
{
    // Every line has number of integers equal to line number
    for (i = 0; i <= line; i++)
    {
    // First and last values in every row are 1
    if (line == i || i == 0)
        arr[line][i] = 1;
    else
    // Other values are sum of values just above and left of above
        arr[line][i] = arr[line-1][i-1] + arr[line-1][i];
    document.write(arr[line][i]);
    }
    document.write("<br>");
}
}
 
// This code is contributed by 29AjayKumar
 
</script>


PHP




<?php
// PHP program for Pascal’s Triangle
// A O(n^2) time and O(n^2) extra space
// method for Pascal's Triangle
function printPascal($n)
{
    // An auxiliary array to store
    // generated pascal triangle values
    $arr = array(array());
     
    // Iterate through every line and
    // print integer(s) in it
    for ($line = 0; $line < $n; $line++)
    {
        // Every line has number of integers
        // equal to line number
        for ($i = 0; $i <= $line; $i++)
        {
            // First and last values in every row are 1
            if ($line == $i || $i == 0)
                $arr[$line][$i] = 1;
                 
            // Other values are sum of values just
            // above and left of above
            else
                $arr[$line][$i] = $arr[$line - 1][$i - 1] +
                                $arr[$line - 1][$i];
            echo $arr[$line][$i] . " ";
        }
        echo "\n";
    }
}
 
// Driver code
$n = 5;
printPascal($n);
 
// This code is contributed
// by Akanksha Rai
?>


Output

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

Time Complexity: O(N^2)
Auxiliary Space: O(N^2)

Note: This method can be optimized to use O(n) extra space as we need values only from previous row. So we can create an auxiliary array of size n and overwrite values. Following is another method uses only O(1) extra space.

Pascal’s Triangle using Binomial Coefficient (Space Optimised):

This method is based on approach using Binomial Coefficient. We know that ith entry in a line number line is Binomial Coefficient C(line, i) and all lines start with value 1. The idea is to calculate C(line, i) using C(line, i-1). It can be calculated in O(1) time.

  • C(line, i) = line! / ( (line-i)! * i! )
  • C(line, i-1) = line! / ( (line – i + 1)! * (i-1)! )
  • We can derive following expression from above two expressions.
  • C(line, i) = C(line, i-1) * (line – i + 1) / i
  • So C(line, i) can be calculated from C(line, i-1) in O(1) time

below is the implementation of the approach:

C++




// C++ program for Pascal’s Triangle
// A O(n^2) time and O(1) extra space
// function for Pascal's Triangle
#include <bits/stdc++.h>
 
using namespace std;
void printPascal(int n)
{
 
    for (int line = 1; line <= n; line++) {
        int C = 1; // used to represent C(line, i)
        for (int i = 1; i <= line; i++) {
 
            // The first value in a line is always 1
            cout << C << " ";
            C = C * (line - i) / i;
        }
        cout << "\n";
    }
}
 
// Driver code
int main()
{
    int n = 5;
    printPascal(n);
    return 0;
}
 
// This code is contributed by Code_Mech


C




// C program for Pascal’s Triangle
// A O(n^2) time and O(1) extra space
// function for Pascal's Triangle
void printPascal(int n)
{
for (int line = 1; line <= n; line++)
{
    int C = 1; // used to represent C(line, i)
    for (int i = 1; i <= line; i++)
    {
    printf("%d ", C); // The first value in a line is always 1
    C = C * (line - i) / i;
    }
    printf("\n");
}
}
// Driver code
int main()
{
int n = 5;
    printPascal(n);
    return 0;
}


Java




// Java program for Pascal's Triangle
// A O(n^2) time and O(1) extra
// space method for Pascal's Triangle
import java.io.*;
class GFG {
 
//Pascal function
public static void printPascal(int n)
{
    for(int line = 1; line <= n; line++)
    {
         
    int C=1;// used to represent C(line, i)
    for(int i = 1; i <= line; i++)
    {
        // The first value in a line is always 1
        System.out.print(C+" ");
        C = C * (line - i) / i;
    }
    System.out.println();
    }
}
 
// Driver code
public static void main (String[] args) {
    int n = 5;
    printPascal(n);
}
}
// This code is contributed
// by Archit Puri


Python3




# Python3 program for Pascal's Triangle
# A O(n^2) time and O(1) extra
# space method for Pascal's Triangle
 
# Pascal function
def printPascal(n):
 
    for line in range(1, n + 1):
        C = 1; # used to represent C(line, i)
        for i in range(1, line + 1):
             
            # The first value in a
            # line is always 1
            print(C, end = " ");
            C = int(C * (line - i) / i);
        print("");
 
# Driver code
n = 5;
printPascal(n);
 
# This code is contributed by mits


C#




// C# program for Pascal's Triangle
// A O(n^2) time and O(1) extra
// space method for Pascal's Triangle
using System;
class GFG
{
 
// Pascal function
public static void printPascal(int n)
{
    for(int line = 1;
            line <= n; line++)
    {
         
    int C = 1;// used to represent C(line, i)
    for(int i = 1; i <= line; i++)
    {
        // The first value in a
        // line is always 1
        Console.Write(C + " ");
        C = C * (line - i) / i;
    }
    Console.Write("\n") ;
    }
}
 
// Driver code
public static void Main ()
{
    int n = 5;
    printPascal(n);
}
}
 
// This code is contributed
// by ChitraNayal


Javascript




<script>
 
// JavaScript program for Pascal's Triangle
// A O(n^2) time and O(1) extra
// space method for Pascal's Triangle
 
//Pascal function
function printPascal(n)
{
    for(line = 1; line <= n; line++)
    {
         
    var C=1;// used to represent C(line, i)
    for(i = 1; i <= line; i++)
    {
        // The first value in a line is always 1
        document.write(C+" ");
        C = C * (line - i) / i;
    }
    document.write("<br>");
    }
}
 
// Driver code
var n = 5;
printPascal(n);
 
// This code is contributed by 29AjayKumar
 
</script>


PHP




<?php
// PHP program for Pascal's Triangle
// A O(n^2) time and O(1) extra
// space method for Pascal's Triangle
 
// Pascal function
function printPascal($n)
{
    for($line = 1; $line <= $n; $line++)
    {
        $C = 1;// used to represent C(line, i)
        for($i = 1; $i <= $line; $i++)
        {
            // The first value in a
            // line is always 1
            print($C . " ");
            $C = $C * ($line - $i) / $i;
        }
        print("\n");
    }
}
 
// Driver code
$n = 5;
printPascal($n);
 
// This code is contributed by mits
?>


Output

1 
1 1 
1 2 1 
1 3 3 1 
1 4 6 4 1 

Time Complexity: O(n2)
Auxiliary Space: O(1)

Variations of the problem that may be asked in interviews:

  • Find the whole pascal triangle as shown above.
  • Find just the one element of a pascal’s triangle given row number and column number in O(n) time.
  • Find a particular row of pascal’s triangle given a row number in O(n) time.
     


Last Updated : 20 Oct, 2023
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