Marathon Prizes

Date: 01/10/98 at 22:15:49
From: Thomas L
Subject: Permutations

Ten runners are competing in a marathon. In how many ways can the 
first and second prizes be awarded? 

Also, There are 7 possible commercials to be used in 3 time slots. How
many possible arrangements are there? 

Thanks for your help.

Date: 01/11/98 at 08:50:07
From: Doctor Anthony
Subject: Re: Permutations

There are 10 ways that the first prize could be awarded, and now with 
9 to choose from, there are 9 ways that the second prize could be 
awarded. So the total number of ways that the two prizes could be 
awarded is 

               10 x 9 = 90 ways.

For the second question we must choose 3 from the 7 to go into the 
time slots.

We could choose the first in 7 ways, then the second in 6 ways and 
finally the third in 5 ways. This would give 7 x 6 x 5 = 210 ways.  
However the order in this question does not matter, and for any 
group of 3 commercials, we could rearrange them amongst themselves 
in 3 x 2 x 1 = 6 ways. So the group of 3 commercials could be chosen 
in 210/6  = 35 ways.


In the first question we were dealing with permutations where the 
order matters, while in the second question we were dealing with 
combinations where order does not matter.

There are formulae that can be used in each situation.  

  nPr is the number of permutations of r things that can be made from 
n different things. 10 P 2 = 10 x 9 = 90 = answer to first question.

  nCr is number of combinations of r things that can be made from n 
different things

               7 x 6 x 5      210
      7 C 3 =  ---------   = -----  = 35  = answer to second question.
               1 x 2 x 3       6 

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/