Pascal's triangle is a pattern of the triangle which is based on nCr, below is the pictorial representation of Pascal's triangle.

Example:

Input: N = 5
Output:
      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1

Method 1: Using nCr formula i.e. n!/(n-r)!r!

After using nCr formula, the pictorial representation becomes:

          0C0
       1C0   1C1
    2C0   2C1   2C2
 3C0   3C1   3C2    3C3

Algorithm:

• Take a number of rows to be printed, lets assume it to be n
• Make outer iteration i from 0 to n times to print the rows.
• Make inner iteration for j from 0 to (N - 1).
• Print single blank space " ".
• Close inner loop (j loop) //its needed for left spacing.
• Make inner iteration for j from 0 to i.
• Print nCr of i and j.
• Close inner loop.
• Print newline character (\n) after each inner iteration.

Implementation:

# Print Pascal's Triangle in Python
from math import factorial

# input n
n = 5
for i in range(n):
    for j in range(n-i+1):

        # for left spacing
        print(end=" ")

    for j in range(i+1):

        # nCr = n!/((n-r)!*r!)
        print(factorial(i)//(factorial(j)*factorial(i-j)), end=" ")

    # for new line
    print()

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

Time complexity: O(N²)
Auxiliary space: O(1)

Method 2: We can optimize the above code by the following concept of a Binomial Coefficient, the i’th 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).

C(line, i) = C(line, i-1) * (line - i + 1) / i

Implementations:

# Print Pascal's Triangle in Python

# input n
n = 5

for i in range(1, n+1):
    for j in range(0, n-i+1):
        print(' ', end='')

    # first element is always 1
    C = 1
    for j in range(1, i+1):

        # first value in a line is always 1
        print(' ', C, sep='', end='')

        # using Binomial Coefficient
        C = C * (i - j) // j
    print()

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

Time complexity: O(N²)
Auxiliary Space: O(1)

Method 3: The code prints Pascal's Triangle up to the 6th row. It iterates through each row and calculates each value using the binomial coefficient formula, which is

\frac{n!}{k! (n-k)!}

where n is the row number and k is the position in the row.

Implementation:

# Print Pascal's Triangle in Python

# input n
n = 6

# iterate up to n
for i in range(n):
    # adjust space
    print(' '*(n-i), end='')

    # compute each value in the row
    coef = 1
    for j in range(0, i + 1):
        print(coef, end=' ')
        coef = coef * (i - j) // (j + 1)
    print()

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

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