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
Using nCr formula i.e. n!/(nr)!r!
After using nCr formula, the pictorial representation becomes:
0C0
1C0 1C1
2C0 2C1 2C2
3C0 3C1 3C2 3C3
Algorithm:
Implementation:
# Print Pascal's Triangle in Python
from math import factorial
# input n
n = 5
for i in range(n):
for j in range(ni+1):
# for left spacing
print(end=" ")
for j in range(i+1):
# nCr = n!/((nr)!*r!)
print(factorial(i)//(factorial(j)*factorial(ij)), end=" ")
# for new line
print()
Output:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
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, i1).
C(line, i) = C(line, i1) * (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, ni+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
This is the most optimized approach to print Pascal’s triangle, this approach is based on powers of 11.
11**0 = 1
11**1 = 11
11**2 = 121
11**3 = 1331
Implementation:
# Print Pascal's Triangle in Python
# input n
n = 5
# iterarte upto n
for i in range(n):
# adjust space
print(' '*(ni), end='')
# compute power of 11
print(' '.join(map(str, str(11**i))))
Output:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
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49 videos38 docs18 tests


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