The binomial coefficients can be arranged in rows to form Pascal’s Triangle (where row n is (n 0),(n 1),...(n n). In which row of Pascal’s Triangle do three consecutive entries occur that are in the ratio 3:4:5?

binomial.py

#!/usr/bin/env python3
#-*- coding: utf-8 -*-

from __future__ import print_function

def binomial(n, k):

	if n < 0 or k < 0 or k > n:
		return 0

	if n == k:
		return 1

	result = n

	for i in range(1, k):
		result *= n - i
		result //= i + 1

	return result

def solve_the_problem(maxn = 100):
	for n in range(maxn + 1):
		for k in range(1, n >> 1):
			a, b, c = binomial(n, k - 1), binomial(n, k), binomial(n, k + 1)
			if 4*a == 3*b and 5*b == 4*c:
				return n, k

n, k = solve_the_problem()
print(n, k)