Problem 53

Solution to Problem 53. Written in Python.

Problem: 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, ⁿC_r = 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 ⁿC_r, for 1 ≤ n ≤ 100, are greater than one-million?

from math import factorial

nCr = lambda n, r: factorial(n) / factorial(r) / factorial(n - r)
ncr_list = [nCr(n, r) for n in range(1, 101) for r in range(1, n + 1)]
print len([i for i in ncr_list if i > 1000000])