TURKEYS puzzle

Date: 11/28/2001 at 19:00:54
From: Donna
Subject: Math puzzle

To solve this puzzle, count the number of ways you can trace the word 
"TURKEYS" in the triangular array. Only one rule, you may only move to 
one of the two letters directly below the letter you are on.
 
                                  T
                                 U U
                                R R R
                               K K K K 
                              E E E E E 
                             Y Y Y Y Y Y 
                            S S S S S S S 

I know there is a math formula or equation but I'm not sure what.
First I thought maybe multiply the number of letters times each other
but I'm not sure. Could you please help? Thank you so much!

Date: 11/29/2001 at 12:54:42
From: Doctor Greenie
Subject: Re: Math puzzle

Hello, Donna -

When you go from the top T row to the next U row, you have two choices 
of which way to go.

When you go from one of the entries in the U row to the next R row, 
you have two choices of which way to go, regardless of which U you 
start from. This means that, for each of the two different ways you 
had to get from the start to the U row, you have two choices for going 
from the U row to the R row.  That makes 2*2 = 4 different ways you 
can get from the start to the R row.

When you go from one of the entries in the R row to the next K row, 
you have two choices of which way to go, regardless of which R you 
start from.  This means that, for each of the 2*2 = 4 different ways 
you had to get from the start to the R row, you have two choices for 
going from the R row to the K row. That makes 2*2*2 = 8 different ways 
you can get from the start to the K row.

So what do you think the pattern is here? And so how many different 
ways will there be for you to get from the top of the triangle to the 
bottom?

I hope this helps. Write back if you have any further questions on 
this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/