Python program to print Pascal’s Triangle
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()
Output
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()
Output
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
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()
Output
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)