Pascal's
triangle  a code with forloops in Matlab
The
Pascal's
triangle is a triangular array of the binomial coefficients. The
binomial
coefficients appear as the numbers of Pascal's triangle. We can
calculate the
elements of this triangle by using simple iterations with Matlab. The
idea is
to practice our forloops and use our logic.
In
algebra, the binomial theorem describes the expansion of powers of a
binomial.
According to this theorem, it’s possible to expand the power (x + y)^{n}
into a sum involving terms of the form Ax^{b}y^{c}, where the
exponents b and
c are positive integers with b + c = n,
and the coefficient A of each term is an integer depending on n and b.
For
example, the fourth row in the triangle shows numbers 1 3 3 1, and that
means the
expansion of a cubic binomial, which has four terms.
(x + y)^{3}
= x^{3}
+ 3x^{2}y + 3xy^{2}
+ y^{2}
The
rows of Pascal's triangle are enumerated starting with row r = 1 at the
top.
The numbers in each row are numbered beginning with column c = 1.
An easy
construction of the triangle could be as follows...
 On
row 1, write only the number 1.
 The
first and last elements of every row are also 1. On row 2, we’ll write
just 1
and 1.
 Then,
to calculate the elements from row 3 on (other than the first and last
figures),
we’ll add the number from the previous row before our column and the
number
from the previous row in our column.
For
example, the fifth row has these five terms (compare with the rows mentioned above)
1
4
6
4
1
Let’s
code that in Matlab...
function pt =
pascal_triangle(n)
%
The first two rows are constant
pt(1, 1)
= 1;
pt(2, 1
: 2) = [1 1];
%
If only two rows are requested, then exit
if n <
3
return
end
for r = 3 :
n
% The first element of every
row is always 1
pt(r, 1) = 1;
% Every element is the addition
of the two elements
% on top of it. That means the
previous row.
for c = 2 : r1
pt(r, c) = pt(r1, c1) + pt(r1, c);
end
% The
last element of every
row is always 1
pt(r,
r) = 1;
end
We can
try the function from the command window, just like this, with the
number of
desired rows as an input parameter:
pt =
pascal_triangle(8)
and we
get the 8row matrix displaying the coefficients
pt
=
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
2
1
0
0
0
0
0
1
3
3
1
0
0
0
0
1
4
6
4
1
0
0
0
1
5 10
10
5
1
0
0
1
6 15
20
15
6
1
0
1
7 21
35
35
21
7
1
The
zeros appear naturally when we change the row in our algorithm.
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'Pascals Triangle' to home
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'Pascals Triangle' to Flow Control
