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					Algebra IIA
Unit 4.5
Pascal’s Coefficients, Jimmy




                               15
Algebra IIA
Unit 4.5
Pascal’s Coefficients, Jimmy
Investigation 11: Pascal and the Coefficients
It turns out that the entries in Pascal’s triangle are combinatorial coefficients. In
fact, this is the main reason Pascal’s triangle is important in mathematics.

It is standard practice to refer to the top row of Pascal’s triangle as “row 0,” the
next 'row as “row 1,” and so on. Similarly, we refer to the first number in each row
as “entry 0,” the next as “entry 1,” and so on. The reason for using this numbering
system is that it connects the positions of a number to its meaning as a
combinatorial coefficient.

For example, according to this system, the boxed number 10 shown above is
entry 2 of row 5 of Pascal’s triangle. This fits with the fact that 10 is equal to the
                             
                             5
combinatorial coefficient   which tells you how many different bowls of ice cream you can make with two-scoops
                              ,
                             2
                             
(of different flavors) if there are five flavors altogether.
                                                                                          n 
In general, it can be proved that entry r and row n is the combinatorial coefficient   For instance, the row 1 4 6 4
                                                                                         .
                                                                                        r 
                                                              4  4  4  4      4 
1 is row 4, and it consists of the combinatorial coefficients         and  
                                                                   ,      ,      ,      ,          .
                                                               0 1           2
                                                                    3         4 
                                                                              
                     4  4  4  4  4 
     1. Check that         and   do have the numerical values of 1, 4, 6, 4, 1, and explain the
                          ,      ,      ,      ,
                      0 1           2
                           3     4   
                                                                                
         values in terms of bowls of ice cream.


                              
    2. Use the connection between Pascal’s triangle and combinatorial coefficients to find these numerical values.
                   
                   6                                                                       
                                                                                           9
             a.                                                                  c.    
                   5
                                                                                       5
                                                                                           

                  7                                                                    
                                                                                           10
             b.                                                                  d.    
                4                                                                 6 



                                                                              
    3. One feature of Pascal’s triangle is that each row begins and ends with the number 1. In terms of
                                                         n         n 
         combinatorial coefficients, this means that   and   are both equal to 1, for any value of n.
                                                          0
                                                                   n 

         Explain this feature of Pascal’s triangle in terms of bowls of ice cream or using some other model for
         combinatorial coefficients.
                                                     




                                                                  16
Algebra IIA
Unit 4.5
Pascal’s Coefficients, Jimmy
Jimmy in the City…

   1. Jimmy moved to the city. He is standing
      at the corner of Starting and Point
      streets. He needs to meet his mom who
      is standing on the corner of 2nd St. and
      3rd Ave. He has some time before he
      needs to meet her and decides to figure
      out how many possible paths he can
      take. Count the number of paths he can
      take if he can only go north and east.



   2. Suppose his mom is standing at the
      following locations. Count the number of
      paths he can take only moving north
      and east. Look for patterns!
          a. 5th St. and Starting St.
          b. 4th St. and 1st Ave.
          c. 3rd St. and 2nd Ave.
          d. 2nd St. and 3rd Ave. (look at #1)
          e. 1st St. and 4th Ave.
          f. Point St. and 5th Ave.



   3. Can you find a pattern? Describe in complete sentences what you notice. Verify your
      conjecture by picking another location and counting the number of possible paths.




   4. How many paths are there to 19th St and 3rd Ave? (Assuming the city streets continue
      with the same numbering system)




   5. Describe in words or write a formula that would show how you could calculate the
      number of possible paths to any location in the city (say nth St. and mth Ave.) using
      combinatorics.


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