# Combinatoric selections (Euler Problem 53)¶

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https://projecteuler.net/problem=53

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

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

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

In general,

$nCr = \frac{n!}{r!(n−r)!}$, 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?

In [3]:

    import math

    def combinations_naiv(n, r):
        return math.factorial(n) // (math.factorial(r) * math.factorial(n - r))

    assert(combinations_naiv(5, 3) == 10)
    assert(combinations_naiv(23, 10) == 1144066)
    assert(combinations_naiv(20, 20) == 1)

In [7]:

    count = 0

    for n in range (1, 101):
        for r in range(1, n + 1):
            if combinations_naiv(n, r) > 10**6:
                count += 1

    s = count

In [9]:

    print(s)
    assert(s == 4075)

    4075

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