Associated Topics || Dr. Math Home || Search Dr. Math

### 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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search