Pascal's Triangle: Words instead of NumbersDate: 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/ |
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