SOLUTION: four cards are randomly selected from a standard 52 car deck, what is the probability of getting 4 hearts or 4 cards lower than a 9, counting aces as 1?

Let  be the event "to select 4 hearts"
Let  be the event "to select 4 cards lower than 9"

Then we use the formula



First we calculate 

There are  ways to select  cards from a standard deck of . 

So  is the denominator of .

Now we need to find the numerator of .

There are  ways to select  hearts from  without regard to order.

So 

Next we find , of getting  cards lower than a , counting aces as 1.

There are  card ranks lower than , and there are  suits, so that makes
 cards which are lower than .  

There are therefore  ways to select  cards lower than  from the .

So 

Next we find .

This is when all  cards selected are both lower than  and are also hearts.

There are  ways to select  hearts from the  hearts lower than .

So  

So now we can plug in the formula:



  
 






which reduces to



Edwin