Brute force

A direct approach to determine all distinct values of \( \displaystyle \binom n r \leq 1000000 \) for \( 1 \le n \le 100 \) is as follows:

• Calculate \( \displaystyle \binom n r \) using the formula \( \displaystyle \binom n r = \dfrac{n!}{r!(n-r)!} \).
• Iterate over each \( n \) from \( 1 \) to \( 100 \) and \( r \) from \( 1 \) to \( n \) and count the number of \( \displaystyle \binom n r > 1,000,000 \).

The first task can be simply implemented using the math.factorial function.

From solution1.py:

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

The solution is then simpy implemented by iterating over all \( n \) and \( r \) and counting the number of \( \displaystyle \binom n r \) that are greater than \( 1,000,000 \).

From solution1.py:

def combinatoric_selections():
    return sum(ncr(n, r) > 1000000 for n in range(23, 101) for r in range(1, n))