Combinatoric Selections
project-euler
programming
Some simple observations
1 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, \(\displaystyle \binom 5 3 = 10\).
In general, \(\displaystyle \binom n r = \dfrac{n!}{r!(n-r)!}\), where \(r \le n\), \(n! = n \times (n-1) \times ... \times 3 \times 2 \times 1\), and \(0! = 1\).
It is not until \(n = 23\), that a value exceeds one-million: \(\displaystyle \binom {23} {10} = 1144066\).
How many, not necessarily distinct, values of \(\displaystyle \binom n r\) for \(1 \le n \le 100\), are greater than one-million?