Calculating the total number of five-card hands

Hi,

I'm going to demonstrate how to compute the odds of being dealt specific starting hands in 5-card draw. The first thing that would be interesting to know is the total number of starting hands (with a regular deck containing 52 cards).

You have been dealt a 5-card hand. The cards are dealt face down on the table. The first card you look at can be any of the 52 cards from the deck. Once you've looked at the first card, the next card can be any of the remaining 51 cards from the deck and so on. After multiplying these numbers (52, 51, 50, 49 and 48), you have calculated the number of arranged hands you can be dealt.

Let's say that you've made a royal flush in spades. It doesn't matter if the first card you looked at was the ace or the ten of spades. It's a royal flush regardless of the sequence of the cards. If you know that you've been dealt the royal flush in spades (but have not looked at your hand), any of the five cards you have can be the ace. Once you've determined where the ace is, there are four cards that can be the king and so on. You have to divide the number of arranged hands by the number of ways to arrange five different cards (5*4*3*2*1) to get the total number of hands you can be dealt in five card draw.

Total number (n): n = (52*51*50*49*48)/(5*4*3*2*1) = 2 598 960