In the california state lottery, you must select six numbers from fifty-two numbers to win the big prize. the numbers do not have to be in a particular order. what is the probability that you will win the big prize if you buy one ticket?

C(n,r) = [n!/((n-r)!*r!)]  
.  
n is the number of possibilities and it is 52 in question and r is the number of  
things to be taken at a time which is 6  
.  
if we substitiue  
.  
C(n,r) = [52!/((52-6)!*6!)= 52!/((46)!*6!)]  
.  
52! = 52*51*49*48*47*46*45*44*43*42* and 46! = 46*45*44*43*42*  
.
  
you divide 52! by 46! the result after canceling all the   
common terms is just 52*51*50*49*48*47. The question is reduced to    
.  
C(n,r) = (52*51*50*49*48*47)/6!  
.  
and 6! = 6*5*4*3*2*1  
.  
If you want you can divide the numbers from 6! into the numbers in the numerator to simplify  
things a little or you can just take your calculator and multiply out the numerator and  
then divide that answer by 720 which is what 6! equals.  
.  
If you just multiply out the numerator your calculator should tell you that the answer is   
1.46581344*10^10 and when you divide that by 720 you get 20,358,520. This means that for  
every 20,358,520 lottery tickets sold there is likely to be 1 winner among them. Pretty slim  
odds of your ticket being that one.  
.  

If you have a cheap scientific calculator you might examine it carefully to see if it has  
a key function labeled nCr. If you do it will calculate this combination automatically.  
.  
Just enter 52, then press the nCr function, then enter 2, and press the equal sign.

The answer would be 27 because assuming that each package has 1 bar of soap and 1 tube of toothpaste, after 27 packages they would run out of toothpaste