I have found Problem 53 of Project Euler to be a really interesting problem to work with, because there is a brute force solution and then there is a much more elegant solution if you dive into the mathematics behind the question. The problem reads

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,^{5}C_{3}= 10.

In general,

where,and

It is not untiln= 23, that a value exceeds one-million:^{23}C_{10}= 1144066.

How many, not necessarily distinct, values of≤^{n}C_{r}, for 1≤n100, are greater than one-million?

I will present 2 different solution strategies each with 2 different solutions, so lets jump right into it. Continue reading →