## Combinatoric selections

Published on Friday, 26th September 2003, 06:00 pm; Solved by 52783;
Difficulty rating: 5%

### Problem 53

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, $^{n}C_{r} = \dfrac{n!}{r!(n-r)!}$, 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: $^{23}C_{10} = 1144066$.

How many, not necessarily distinct, values of $^{n}C_{r}$ for $1 \le n \le 100$, are greater than one-million?