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Pascal's Triangle: Words instead of Numbers


Date: 08/28/2001 at 20:08:16
From: Ryan
Subject: Pascal's Triangle using words instead of numbers

                 T
                R R
               I I I
              A A A A
             N N N N N
              G G G G
               L L L
                E E
                 S

We have to find how many time you can read the word TRIANGLES in the 
figure.  We must use Pascal's triangle.  I believe the answer is 31 or 
46, but I am stuck.


Date: 08/29/2001 at 10:56:26
From: Doctor Greenie
Subject: Re: Pascal's Triangle using words instead of numbers

Hi, Ryan -

Let's start at the top of this diagram and replace the letters in a 
row one at a time with the numbers of ways we can use each of those 
letters.  After the first couple of rows the diagram looks like 
this...

       1
      1 1
     I I I
    .......

This diagram indicates that there is only 1 way to use the "T" and 
only one way to use each of the two Rs.

If we now continue this process to the next row, the diagram looks 
like this...

       1
      1 1
     1 2 1
    A A A A
   .........

This diagram indicates that there is only 1 way to use the first or 
last I (you can get to the first I only from the R above and to the 
right of the first I; and you can get to the last I only from the R 
above and to the left of the last I), but there are two ways to get to 
the middle I (either from the R above and to the left or from the R 
above and to the right).

Let's take the process one step further (and then you can continue the 
process from there to the solution to the problem....). After we go 
one more row, the diagram looks like this...

       1
      1 1
     1 2 1
    1 3 3 1
   N N N N N
  ...........

This diagram indicates that (1) there is only one way to get to the 
first A - from the I above and to the right; (2) there are three ways 
to get to the second A - from the I above and to the left, which could 
be reached in only 1 way, or from the I above and to the right, which 
could be reached in 2 different ways; (3) there are three ways to get 
to the third A - from the I above and to the left, which could be 
reached in 2 different ways, or from the I above and to the right, 
which could be reached in only 1 way; and (4) there is only one way to 
get to the last A - from the I above and to the left.

You can probably see by looking at the diagram at this point why you 
were asked to solve the problem by "using Pascal's triangle."

See if you can finish the problem from here. (The answer is 70.)

Write back if you have any further questions on this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Discrete Mathematics
High School Permutations and Combinations

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