



There
are 52 cards in a regular deck.



Say you
are being dealt a poker hand. 


The first
card you get can be any one of the 52 cards. 


That
means the possible number of first cards 


that you
could get is ... 





52






After
you get your first card, there are still 51 cards out there. 


Other
players will get some of them. 


But
since we don't know which ones they get, 


it
works like all of the cards are just out there 


in
the same unknown zone. 


As
far as we know or can tell, 


we
might get any one of them 





That
means the number of combinations 


of
first and second cards that we might get is ... 





52 × 51 = 2,652 





There
are also ... 





50
possible 3rd cards you could get 


49
possible 4th cards you could get 


48
possible 5th cards you could get 





That
makes the number of different ways 


that
you can be dealt a 5 card poker hand is ... 





52
× 51 × 50× 49× 48






That's
all true and wonderful, 


but
math types like to write stuff in as small a space as they
can. 


Here's
what they do this time ... 





There
is a rule in math that says ... 











If
you write the rule backwards, you get ... 











So
what?



How
does that help us with ... 





52
× 51 × 50 × 49 × 48






This
is not a real factorial, 


it
would also need to have the numbers from 47 down to 1 


multiplied
times the numbers we do have. 


All
of those numbers together make ...






47!






If
we want to, we can write 52! as ... 





52 × 51 × 50
× 49 × 48 × 47!






So
we can say ... 











and
that can be written as ... 











Well
that's just great ... 


Almost. 





We
still have one more little problem. 


The
way the numbers work. 


These
hands, which all have the same cards ... 











would
be counted as different hands. 


They
all have the same cards in them, 


but
the cards were dealt in a different order. 


Most
of the time (like now), 


the
order in which the cards are dealt 


does
not matter. 





Just
how many ways can you deal the same 5 cards? 





You
can get any of the 5 as a first card. 


Any
one of the remaining 4 as the second card. 


And
so on. Putting these all together, we get ... 





5 × 4 × 3 × 2
× 1 = 120






Since
most of the time we don't care about the card order. 


Our
formula for the number of possible hands ... 











gives
an answer that is way too big. 


Each
possible hand shows up 5! times. 


To
make the numbers work out right 


we
need to divide the formula by 5!. 


An
easy math way to do that is to put 5! in the denominator. 


That
makes the whole formula ... 











What
we have here is ... 











Example: 





How
many different 7 card hands are there 


in
a regular 52 card deck? 


(We
don't care about the order of the cards) 











Simplifying
this so we can multiply, we get ... 











Multiplying
this out, we get ... 











Doing
the division ... 











So
there are just over 133 million different 7 card hands 


if
we don't care what order we get the cards in. 





Pop
Quiz: 





How
many different 7 card hands are there 


in
a 52 card deck if we DO care about the card order. 








Answer: 


Over
674 billion. 





copyright 2005 Bruce Kirkpatrick 
