problem53.py

"""
Combinatoric Selections
Project Euler Problem #53
by Muaz Siddiqui

There are exactly ten ways of selecting three from five, 12345:

123, 124, 125, 134, 135, 145, 234, 235, 245, and 345

In combinatorics, we use the notation, 5C3 = 10.

In general, nCr =   n! / r!(n−r)! ,where r ≤ n, n! = n×(n−1)×...×3×2×1, and 0! = 1.
It is not until n = 23, that a value exceeds one-million: 23C10 = 1144066.

How many, not necessarily distinct, values of  nCr, for 1 ≤ n ≤ 100, are greater 
than one-million?
"""
from euler_helpers import timeit, memoize

@memoize
def factorial(n):
    if n == 0:
        return 1
    else:
        return n * factorial(n-1)

def choose(n, r):
    return factorial(n) / (factorial(r) * factorial(n-r))

@timeit
def answer():
    count = 0
    for n in range(1, 101):
        for r in range(n, 1, -1):
            if choose(n, r) > 1000000:
                count += 1
    return count