Q:

Can't understand one chance in R of winning where R is some result of factorials.

In lotto game, let you select six no. from 51 no. on a card and the Lotto managers pick six no. at random. If your choice exactly matches theirs, you win a few dollars.
If you have to pick 6 values out of 51, you have 1 chance in R of winning where R is given by:
$$R=\dfrac{51\cdot 50\cdot 49\cdot 48\cdot 47\cdot 46}{~~6\cdot ~~5\cdot ~~4\cdot ~~3\cdot ~~2\cdot ~~1}$$
What is R and what is this formula? And what do you mean by statement "one chance in R"?

A:

You're looking at the binomial coefficient, written as 
$\dbinom{51}{6}$, said "$51$ choose $6$" and compactly represented as :
$$\binom{51}{6} = \frac{51!}{6!(51-6)!} \\[5ex]
= \frac{51\times 50\times 49\times 48\times 47\times 46}{6\times 5\times 4\times 3\times 2\times 1} \\[5ex]
= 17\times 10\times 49\times 47\times 46 =18009460
$$
This is the total number of ways of selecting a smaller set ($6$) from a larger ($51$). Thus to get a particular desired selection, there is one way to do that in this large number of possibilities.

