4.5 Applying Pascal's Method

The iterative process that generates the terms in Pascal's triangle can also be appliedto counting paths or routes between two points. Pascal's triangle has many uses in binomial expansions. For example:

(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2.
Notice the coefficients are the third row of Pascal's triangle: 1, 2, 1. In general, when a binomial is raised to a positive integer power we have:

(x + y)n = a0xn + a1xn−1y + a2xn−2y2 + … + an−1xyn−1 + anyn,
where the coefficients ai in this expansion are precisely the numbers on row n + 1 of Pascal's triangle. Thus:

This is the binomial theorem.

In order to prove this theorem, one must start by considering that the entire right diagonal corresponds to the coefficient of x0 when (x + 1) has been raised to some power equal to the row number. The next diagonal corresponds to the coefficient of x1, and so on.

Some simple patterns are immediately apparent in Pascal's triangle:

The diagonals going along the left and right edges contain only 1's.
The diagonals next to the edge diagonals contain the natural numbers in order.
Moving inwards, the next pair of diagonals contain the triangular numbers in order.
The next pair of diagonals contain the tetrahedral numbers in order, and the next pair give pentatope numbers.

TO BE CONTINUED……. Example 1

Determine how many different paths will spell PASCALif you start at the top and proceed to the next row by moving diagonally left or right

P
A A
S S S
C C C C
A A A
L L
Solution

Starting at the top record the number of possible paths moving diagonally to the left and right as you proceed to each different letter. For instance there is one path from P to the left A and one path from P to the right A. There is one path from an A to the left S, two paths from an A to the middle S, and one path from an A to the right S. Continuing with this counting reveals that there are 10 different paths leading to each L. Therefore, a total of 20 paths spell PASCAL.
1
P
1 1
A A
1 2 1
S S S
1 3 3 1
C C C C
4 6 4
A A A
10 10
L L

Comment:
By: Osman Osman
Nicely done ….. I like this huge Pascal Triangle that u posted